Slow-light fiber bragg grating sensor

ABSTRACT

In certain embodiments, an optical device and a method of use is provided. The optical device can include a fiber Bragg grating and a narrowband optical source. The narrowband optical source can be configured to generate light. A first portion of light can be transmitted along a first optical path extending along and through the length of the fiber Bragg grating at a group velocity. The light can have a wavelength at or in the vicinity of a wavelength at which one or more of the following quantities is at a maximum value: (a) the product of the group index spectrum and a square root of the power transmission spectrum, (b) the slope of a product of the group index spectrum and one minus the power transmission spectrum, and (c) the slope of a product of the group index spectrum and the power transmission spectrum.

CLAIM OF PRIORITY

The present application claims the benefit of priority to U.S.Provisional Patent Application No. 61/381,032, filed on Sep. 8, 2010,and incorporated in its entirety by reference herein.

RELATED APPLICATIONS

This application is related to U.S. patent application Ser. No.12/792,631, filed Jun. 2, 2010, which is incorporated in its entirety byreference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This application relates generally to optical devices utilizing fiberBragg gratings and slow light, and more particularly, to optical sensorsutilizing fiber Bragg gratings and slow light.

2. Description of the Related Art

Fiber Bragg gratings (FBGs) are used extensively in research and inindustry for a large number of photonics applications, in particular incommunication systems, in fiber lasers, and in fiber sensors. They areused as filters, high or partial reflectors, dispersion compensators,frequency standards, frequency stabilizers, spectrum analyzers, etc. Inthe field of fiber sensors, which is the main area germane to certainembodiments described herein, FBGs are used to sense changes to a numberof perturbations applied individually or simultaneously to the FBG,mostly strain and temperature. For example, when a temperature change isapplied to an FBG, three of the FBG parameters change, namely its length(through thermal expansion) and therefore the period of the grating, theeffective index of the mode propagating in the core (through thethermo-optic effect), and the dimension of the fiber core (again throughthermal expansion). Of these three effects, the one with the largestcontribution to the performance of the FBG is typically the thermo-opticeffect. Combined, these three changes result in a change in the Braggwavelength, which can be measured to recover the temperature changeapplied to the grating. A similar principle is commonly used to measurea longitudinal strain applied to an FBG: when the fiber is strained, thethree parameters mentioned above also change, which causes a shift inthe Bragg wavelength. FBGs are undoubtedly the most widely used opticalsensing component in the field of fiber sensors, largely because oftheir compactness, their ease of manufacturing, and their relativestability, considering that they are, after all, a very sensitivemulti-wave interferometer.

SUMMARY

In certain embodiments, an optical device comprises a fiber Bragggrating comprising a substantially periodic refractive index modulationalong a length of the fiber Bragg grating. The fiber Bragg grating has apower transmission spectrum as a function of wavelength comprising aplurality of local transmission maxima. The local transmission maximaeach have a maximum power at a transmission peak wavelength. The fiberBragg grating has a group index spectrum as a function of wavelength.The device further comprises a narrowband optical source in opticalcommunication with a first optical path and a second optical path. Thenarrowband optical source is configured to generate light. The device isconfigured to split the light into a first portion and a second portion,the first portion transmitted along the first optical path extendingalong and through the length of the fiber Bragg grating at a groupvelocity. The light has a wavelength at or in the vicinity of awavelength at which one or more of the following quantities is at amaximum value: (a) the product of the group index spectrum and a squareroot of the power transmission spectrum, (b) the slope of a product ofthe group index spectrum and one minus the power transmission spectrum,and (c) the slope of a product of the group index spectrum and the powertransmission spectrum.

In certain embodiments, a method of using a fiber Bragg gratingcomprises providing a fiber Bragg grating comprising a substantiallyperiodic refractive index modulation along a length of the fiber Bragggrating. The fiber Bragg grating has a power transmission spectrum as afunction of wavelength comprising a plurality of local transmissionmaxima. The local transmission maxima each have a maximum power at atransmission peak wavelength. The fiber Bragg grating has a group indexspectrum as a function of wavelength. The method further comprisesgenerating light from a narrowband optical source. The narrowbandoptical source is in optical communication with a first optical path anda second optical path, wherein the light is split into a first portionand a second portion. The method further comprises transmitting thefirst portion of light along the first optical path extending along andthrough the length of the fiber Bragg grating at a group velocity. Thelight has a wavelength at or in the vicinity of a wavelength at whichone or more of the following quantities is at a maximum value: (a) theproduct of the group index spectrum and a square root of the powertransmission spectrum, (b) the slope of a product of the group indexspectrum and one minus the power transmission spectrum, and (c) theslope of a product of the group index spectrum and the powertransmission spectrum.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic of a generic apparatus used to measure theshift in Bragg wavelength.

FIG. 2 shows a diagram of a generic implementation of signal processingutilizing a Mach-Zehnder interferometer with a fiber Bragg gratingsensor.

FIG. 3 illustrates the coherence length of light reflected by an FBG asa function of the FBG's index contrast for three different FBG lengthsfor a wavelength of 1.55 microns, calculated assuming a losslessgrating.

FIG. 4 illustrates the calculated maximum sensitivity to temperature ofthe Bragg-reflection-mode FBG sensor of FIG. 2 as a function of indexcontrast for various FBG lengths for a wavelength of 1.55 microns,calculated assuming a lossless grating.

FIG. 5 illustrates the calculated maximum sensitivity to temperature ofthe Bragg-reflection-mode FBG sensor of FIG. 2 as a function of FBGlengths for various index contrasts for a wavelength of 1.55 microns,calculated assuming a lossless grating.

FIG. 6A is a schematic of an optical sensor in accordance with certainembodiments described herein.

FIG. 6B is a schematic of an optical sensor in accordance with certainembodiments described herein.

FIG. 7 shows a diagram of an example implementation of an apparatusutilizing an FBG used in the slow-light transmission mode in accordancewith certain embodiments described herein.

FIG. 8 shows a diagram of an example implementation of an apparatusutilizing an FBG used in the slow-light reflection mode in accordancewith certain embodiments described herein.

FIGS. 9A and 9E illustrate the calculated power transmission spectrumfor an example FBG used in transmission for wavelengths of 1.064 micronsand 1.55 microns, respectively, in accordance with certain embodimentsdescribed herein, calculated assuming a lossless grating. (The“unheated” curve shows the case without thermal perturbation, the“heated” curve with thermal perturbation.)

FIGS. 9B and 9F illustrate the calculated transmitted signal phase as afunction of wavelength for an example FBG used in transmission forwavelengths of 1.064 microns and 1.55 microns, respectively, inaccordance with certain embodiments described herein, calculatedassuming a lossless grating.

FIGS. 9C and 9G illustrate the calculated group index as a function ofwavelength for an example FBG used in transmission for wavelengths of1.064 microns and 1.55 microns, respectively, in accordance with certainembodiments described herein, calculated assuming a lossless grating.

FIGS. 9D and 9H illustrate the calculated temperature sensitivity for anexample FBG used in transmission for wavelengths of 1.064 microns and1.55 microns, respectively, in accordance with certain embodimentsdescribed herein, calculated assuming a lossless grating.

FIGS. 10A and 10B illustrate the relationship between phase sensitivityto temperature as a function of power transmission for an example FBGused in the transmission mode for λ_(Bragg)=1.064 μm and forλ_(Bragg)=1.55 μm, respectively, in accordance with certain embodimentsdescribed herein in the vicinity of slow-light wavelengths, calculatedassuming a lossless grating.

FIGS. 11A and 11B illustrate the relationship between power sensitivityto temperature as a function of index contrast for FBGs of fixed length(e.g., 2 cm) used in the slow-light transmission mode (dashed line) forλ_(Bragg)=1.064 μm and for λ_(Bragg)=1.55 μm, respectively, inaccordance with certain embodiments described herein, in the slow-lightreflection mode (dotted line) in accordance with certain embodimentsdescribed herein, and in the Bragg-reflection mode (solid line),calculated assuming a lossless grating.

FIGS. 12A and 12B illustrate the group index calculated as a function ofindex contrast Δn for an FBG of fixed length (2 cm) used in theslow-light reflection mode (solid line) and the slow-light transmissionmode (dashed line) for λ_(Bragg)=1.064 μm and for λ_(Bragg)=1.55 μm,respectively, in accordance with certain embodiments described herein,calculated assuming a lossless grating.

FIGS. 13A and 13B illustrate the relationship between the powersensitivity to temperature as a function of length for an FBG of fixedindex contrast (1.5×10⁻⁴) for the slow-light reflection mode (dottedline) and the slow-light transmission mode (dashed line) forλ_(Bragg)=1.064 μm and for λ_(Bragg)=1.55 μm, respectively, inaccordance with certain embodiments described herein, and the Braggreflection mode (solid line), calculated assuming a lossless grating.

FIGS. 14A and 14B illustrate the calculated minimum detectabletemperature change for an FBG of 1 cm in length for λ_(Bragg)=1.064 μmand 2 cm in length for λ_(Bragg)=1.55 μm used in the Bragg-reflectionmode as a function of index contrast Δn, calculated assuming a losslessgrating.

FIGS. 15A and 15B illustrate the calculated minimum detectabletemperature change for an FBG of 1 cm in length for λ_(Bragg)=1.064 μmand 2 cm in length for λ_(Bragg)=1.55 μm used in the slow-lighttransmission mode in accordance with certain embodiments describedherein as a function of index contrast Δn, calculated assuming alossless grating.

FIG. 16 shows the power sensitivity dependence of the linewidth of thelaser for an FBG of 2 cm in length and Δn=1.5×10⁻⁴ in the slow-lightreflection mode (solid line) and the slow-light transmission mode(dashed line) in accordance with certain embodiments described herein,calculated assuming a lossless grating.

FIG. 17 shows the group index and power transmission as functions oflength for different losses in an example of a strong uniform FBG(Δn=1.0×10⁻³).

FIG. 18 shows the group index and power transmission as functions oflength for a loss of 2 m⁻¹ in an example of a hydrogen-loaded FBG.

FIG. 19A shows the index profiles for FBGs with a uniform profile, aType A apodized profile, and a Type B apodized profile.

FIG. 19B shows the group index spectrum of a type A apodized grating.

FIG. 19C shows a plot of an asymmetric spectrum of power for an exampleFBG.

FIG. 19D shows a plot of an asymmetric spectrum of group index for theexample FBG used in FIG. 19C.

FIG. 20 shows the group index and power transmission as functions oflength for different losses in an example of a strong apodized FBG oftype B.

FIG. 21 shows the group index and power transmission as functions of theFWHM of Gaussian apodization in an example of a strong apodized FBG oftype B.

FIG. 22 shows the group, index and power transmission as functions oflength in an example of a hydrogen-loaded FBG.

FIG. 23 shows the group index and power transmission as functions of theFWHM of Gaussian apodization in the example of a hydrogen-loaded FBG ofFIG. 22.

FIG. 24 shows an example experimental setup used to measure the groupdelay of an FBG.

FIG. 25A shows the measured and theoretical transmission spectra of anexample FBG.

FIG. 25B shows the measured and theoretical group index spectra for thesame example FBG used in FIG. 25A.

FIGS. 26A shows the full measured transmission spectrum for an exampleFBG.

FIG. 26B shows the short-wavelength portion of the measured andtheoretical transmission spectra shown in FIG. 26A.

FIG. 26C shows the measured and theoretical group index spectra for theexample FBG used in FIG. 26A.

FIGS. 27-28 are flowcharts of example methods for optically sensing inaccordance with certain embodiments described herein.

FIG. 29A shows the transmission spectrum of a π-shifted grating.

FIG. 29B shows the group index spectrum of a π-shifted grating used inFIG. 29A.

FIG. 30 shows an example transmission spectrum calculated and thecalculated group index spectrum for the example apodized grating for anexample apodized grating.

FIG. 31 shows a diagram of an example implementation of an apparatusutilizing an FBG used in the slow-light transmission mode in accordancewith certain embodiments described herein.

FIG. 32 shows an example figure of merit, which is the product of thesquare root of the transmission spectrum by the group index spectrum ofFIG. 30.

FIGS. 33A-33C are flowcharts of example methods for optically sensing inaccordance with certain embodiments described herein.

FIG. 34A shows a measured transmission spectrum for an example FBG.

FIG. 34B shows a group index spectrum for the example FBG used in FIG.36A.

FIG. 35 shows an example experimental setup utilizing an FBG in aMach-Zehnder interferometer to test its performance as a slow-lightsensor.

FIG. 36 shows the sensitivity measured at the four slow light peaksmeasured in the example experimental setup of FIG. 35.

FIG. 37 shows the FWHM bandwidth as a function of the index contrast inan example uniform grating with a length L=2 cm and no loss.

FIG. 38 illustrates the relationship between sensitivity to strain as afunction of index contrast for an FBG with a length L=2 cm and no lossutilized in the slow-light transmission mode (upper solid line), inaccordance with certain embodiments described herein, with loss utilizedin the slow-light transmission mode (various dotted and dashed lines),in accordance with certain embodiments described herein, and theconventional reflection mode with Mach-Zehnder (MZ) processing (lowersolid line).

DETAILED DESCRIPTION

Although fiber Bragg gratings (FBGs) can take many forms that differ intheir details, an FBG typically includes of a periodic index grating ofperiod Λ fabricated along the guiding region of an optical fiber. Thepresence of a periodic structure in the waveguiding region of an FBGinduces a photonic bandgap, namely a band of finite bandwidth in theoptical frequency space where light is not allowed to propagate forwardthrough the grating. The central wavelength of this bandgap is known asthe Bragg wavelength, λ_(Bragg). When light of wavelengths in thevicinity of λ_(Bragg) is injected into the core of an FBG, it issubstantially reflected from the FBG, while light of wavelengthssufficiently far away from λ_(Bragg) is substantially transmitted alongthe length of the FBG. A physical explanation for this reflection isthat each ripple in the index of the core region reflects a smallfraction of the incident light into the backward-propagating fundamentalmode of the fiber. This reflection is physically due to Fresnelreflection occurring at the interface between two dielectric media ofdifferent refractive indices. The fraction of light (in terms ofelectric field) that is reflected at each ripple is thereforeproportional to Δn, which is a very small number. However, an FBGtypically contains tens of thousands of periods, so all thesereflections can add up to a sizeable total reflection. At the Braggwavelength, the grating period Λ is such that substantially all theindividual reflections are in phase with each other. All reflectionsthen add constructively into the backward-propagating mode, which canend up carrying a large fraction of the incident light's power. In anFBG with a sufficiently long length and strong index modulation Δn,essentially 100% of the incident light can be reflected

In the field of fiber sensors, most FBGs to date have been used in whatis referred to herein as the Bragg-reflection mode. A schematic of thismode of operation is shown in FIG. 1. Light (for example from abroadband light source) is launched into the FBG through a fibercoupler. The portion of the light spectrum centered around λ_(Bragg)that is reflected from the FBG is split by the same coupler and directedtoward a wavelength-monitoring instrument, for example an opticalspectrum analyzer (OSA), which measures λ_(Bragg). Alternatively, theportion of the light spectrum that is transmitted by the FBG can bemeasured by a second wavelength-monitoring instrument, again for examplean OSA, which also provides a measured value of λ_(Bragg). When atemperature change is applied to the FBG, λ_(Bragg) changes, this changeof λ_(Bragg) (or the changed value of λ_(Bragg)) is measured by one orboth of the wavelength-monitoring instruments, and the absolute value ofthe temperature change can then be calculated from the measured changeof λ_(Bragg) (or the changed value of λ_(Bragg)). The same principle isused to measure the absolute (or relative) magnitude of any otherperturbation applied to the FBG that modifies λ_(Bragg), such as astrain an train or acceleration. Many examples of this mode of operationof FBGs as sensors are described in the literature. All of them havethis point in common that they rely on a measurement of the Braggwavelength λ_(Bragg) (or the changed value of λ_(Bragg)) to recover themeasurand.

In order to improve the sensitivity of an FBG used in theBragg-reflection mode, it is essential to improve the ability to measureextremely small changes in wavelength, e.g., changes of less than 10⁻¹³meters. This can be accomplished by utilizing an OSA with a highresolution. Commercial OSAs are available with a sufficiently highwavelength resolution. For example, Yokogawa Electric Company of Tokyo,Japan markets an OSA which has a resolution of 0.05 nm, and AnritsuCorporation of Atsugi, Japan offers an OSA with a resolution of 0.07 nm.

Another solution, which provides a much higher wavelength resolution,e.g., a resolution of 10⁻¹² m, than a conventional OSA, is to use animbalanced Mach-Zehnder (MZ) interferometer to monitor the wavelength.See, e.g., A. D. Kersey, T. A. Berkoff, and W. W. Morey, “Highresolution fibre-grating based strain sensor with interferometricwavelength-shift detection,” Electronic Letters, Vol. 28, No. 3.(January 1992). A diagram of a generic implementation, of this conceptin shown in FIG. 2. The signal reflected from the FBG, which has awavelength λ_(Bragg) to be measured, goes through the two arms of the MZinterferometer. Because the two arms have different length L₁ and L₂,e.g., 50 cm for L₁ and 51 cm for L₂, with proper phase biasing, thesignal coming out of one output port of the MZ interferometer isproportional to sin(Δφ/2), where Δφ=2πnΔL_(Bragg), n is the modeeffective index in the MZ fiber, and ΔL=L₁−L₂. If, as a result of aperturbation applied to the FBG, the Bragg wavelength of the FBG variesby δλ_(Bragg), then the phase difference between the two arms of the MZinterferometer will vary by

|δΔφ|=−2πnΔLδλ _(Bragg)/λ_(Bragg) ²  (1)

With suitable phase bias of the MZ interferometer, the detected power inthe presence of the perturbation is proportional to sin(Δφ/2), and thusit varies by sin(πnΔLδλ_(B)/λ_(B) ²), and δλ_(Bragg) can be recovered bymeasuring this variation in power. For a small perturbation, δλ_(Bragg)is small, and so is δΔφ so the power change is then proportional toΔLδλ_(Bragg)/λ_(Bragg) ². Hence, this technique can give, in principle,a very high resolution in δλ_(Bragg) by increasing ΔL to a very highvalue, which is easy to do because an optical fiber typically has verylow loss (so a long length can be used without the penalty of increasedsignal loss and thus reduced signal-to-noise ratio) and is inexpensive.

The approach of FIG. 2 has two main limitations. The first one is thatthe imbalance ΔL cannot be increased indefinitely. The basic operationof an MZ interferometer requires that the two signals are recombined(e.g., at the second coupler in the MZ interferometer) with a highdegree of temporal coherence, so that they can interfere. This meansthat the optical length mismatch ΔL is selected to not exceedapproximately the coherence length L_(c) of the signal traveling throughthe MZ interferometer. This coherence length is related to the frequencylinewidth Δν (or wavelength linewidth Δλ) of the signal that isreflected by the FBG by:

$\begin{matrix}{L_{c} = {\frac{c}{\pi \; \Delta \; v} = \frac{\lambda^{2}}{\pi \; \Delta \; \lambda}}} & (2)\end{matrix}$

In turn, the linewidth of the light reflected from a grating isapproximately given by:

$\begin{matrix}{{\Delta \; \lambda} = {\lambda_{Bragg}\sqrt{\left( \frac{\Delta \; n}{2n} \right)^{2} + \left( \frac{1}{N} \right)^{2}}}} & (3)\end{matrix}$

where N=L/Λ is the number of periods in the grating, and L is the FBGlength. See, for example, Y. J. Rao, “In-fibre Bragg grating sensors,”Meas. Sci. Technol. Vol. 8, 355-375 (1997). The second condition (anarrow reflected linewidth) can therefore be met by decreasing the indexmodulation of the FBG, and/or increasing the number of periods, e.g.,increasing the length of the FBG.

If the linewidth of the reflected signal is narrow, the signal coherencelength is long, a large length imbalance can be used in the MZinterferometer, and the sensitivity can be high. However, the linewidthof the reflected signal cannot be made arbitrarily narrow. The linewidthis constrained, through Eq. 3, by the grating, namely by the number ofperiods N and the relative index contrast Δn/n. To be able to use alarge path mismatch ΔL, one can use a very weak grating (very smallrelative index contrast (or modulation) Δn/n and a very long grating).For example, to use a 1-m path mismatch at a wavelength of 1.55 μm, acoherence length of 1 m is used or, according to Eq. (3), for example arelative index contrast of ˜10 ⁻⁵ and a grating length greater than 16cm.

FIG. 3 illustrates the coherence length of light reflected from an FBGas a function of its index contrast Δn for three different FBG lengthsfor a wavelength of 1.064 microns, calculated assuming a losslessgrating. For a given FBG length, as the index contrast decreases, thecoherence length increases up to some maximum value. This maximum valueincreases as the grating length is increased. This maximum coherencelength is approximately equal to the grating length (see FIG. 3). Thisresult is expected from Eq. (2) and Eq. (3): in the limit of negligibleΔn/n, Δλ approaches λ_(Bragg)/N, and therefore L_(c) approaches

$\begin{matrix}{L_{c} = {\frac{\lambda_{Bragg}^{2}}{\pi \; \Delta \; \lambda} = {{\frac{\lambda_{Bragg}}{\pi}N} = {{\frac{2n\; \Lambda}{\pi}N} = {\frac{2n}{\pi}L}}}}} & (4)\end{matrix}$

where the expression of the Bragg wavelength of an FBG, λ_(Bragg)=2nΛ,has been used. In a silica fiber, n≈1.45, hence 2n/π in Eq. (4) is equalto 0.92, so L_(c) is close to L, as predicted in FIG. 3. For a givenindex contrast, as the length increases, the coherence length alsoreaches a plateau, beyond which further increasing the FBG length doesnot increase the coherence length in any significant manner. Thus, toget long coherence lengths, one can use a grating with a low contrastand a long length. However, for typical contrast values (10⁻⁴ to 10⁻⁶,10⁻⁵ being about the most typical) even for FBG lengths exceedingtypical reasonable values (a few cm), the coherence length is only ofthe order of 10 cm or less. This is consistent with the report of Kerseyet al., in which a length mismatch of 10 mm was used in the MZinterferometer that processed the reflected signal from an FBG with alinewidth Δλ=0.2 nm (corresponding to a coherence length of ˜3.8 mmaccording to Equation 2).

Based on the foregoing, the sensitivity of the Bragg-reflectionconfiguration of FIG. 2 is limited by the length mismatch, which isitself limited by the coherence length of the reflected signal, whichitself is imposed by Δn and L according to FIG. 3. For a length mismatchΔL, the change in the phase difference Δφ between the two arms of the MZinterferometer resulting from a change in the wavelength δλ_(Bragg)reflected by the FBG is given by Equation 1. Assuming to first orderthat this change in wavelength is primarily due to a change of fiberindex with temperature (e.g., neglecting the effect of the change in FBGlength and fiber transverse dimension), the sensitivity of the sensor ofFIG. 2 can be written explicitly as

$\begin{matrix}{\frac{\partial{\varphi (\lambda)}}{\partial T} = {{\frac{2\pi \; n\; \Delta \; L}{\lambda}\left( {\frac{1}{\lambda}\frac{\partial\lambda}{\partial T}} \right)} \approx {\frac{2\pi \; \Delta \; L}{\lambda}\left( \frac{\partial n}{\partial T} \right)}}} & (5)\end{matrix}$

The sensitivity is a simple linear function of ΔL. For a silica fiber,dn/dT≈1.1×10⁻⁵° C.⁻⁴. For the exemplary maximum length mismatch of 10 cmused in FIG. 3, and for a Bragg wavelength around 1.064 μm, Equation 5states that the phase sensitivity to temperature is about 6.5 rad/° C.If the minimum phase change detectable at the output of the MZinterferometer is 1 μrad (a typical good value), the minimum detectabletemperature change 10⁻⁶/6.5≈1.54×10⁻⁷° C.

The discussion above assumes a certain arm length mismatch of 10 cm(which is applicable, for example, for a grating length of about 10 cmand a contrast below 10⁻⁵, see FIG. 3). In practice, the maximumpossible imbalance is determined by the FBG's index contrast and length,according to FIG. 3, and it is less than 10 cm. FIG. 4 illustrates thesensitivity to temperature plotted as a function of index contrast fordifferent grating lengths for a wavelength of 1.064 microns, calculatedassuming a lossless grating. For a given grating length, the sensitivityincreases as the index contrast decreases, up to an asymptotic maximum.This asymptotic maximum increases as the grating length increases. FIG.4 shows that much smaller sensitivities result when a high contrast isused, even if the length of the device is increased.

FIG. 5 illustrates the sensitivity to temperature plotted as a functionof grating length for different index contrast for a wavelength of 1.064microns, calculated assuming a lossless grating. The phase sensitivityincreases as the length of the grating increases, up to an asymptoticmaximum. To achieve a high sensitivity in the Bragg-reflection mode, theFBG's index contrast can be selected to be low, and its length can beselected to be long. In addition, the maximum practical sensitivity isof the order of 10 rad/° C. (see FIG. 5), or a minimum detectabletemperature of the order of 10⁻⁷° C.). Smaller values can be obtained byfurther reducing the index contrast and increasing the length, but atthe price of a longer device.

The second limitation of the approach of FIG. 2 is that an imbalanced MZinterferometer is highly sensitive to temperature variations, and moreso as the imbalance increases. As the signal propagates through eacharm, it experiences a phase shift proportional to the length of this arm(e.g., φ₁=2πnL₁/λ_(Bragg) for arm 1). If the entire MZ interferometer isinadvertently subjected to a temperature change ΔT, the phase of thesignals in arm 1 and in arm 2 will vary by different amounts, and as aresult the phase bias of the MZ interferometer will change. It isdesirable to not allow this phase bias to vary too much, otherwise thesensitivity of the MZ interferometer will vary over time between theoptimum value (for the optimum bias) and zero. Therefore, it isdesirable to stabilize the temperature of the MZ interferometer. Forlarger length mismatches ΔL, this temperature control is desirably moretight, which is difficult to implement in practice. For example,consider the example of a fiber made of silica, with a signal wavelengthof 1.064 μm, and an arm length mismatch of 10 cm. For the phasedifference between the two arms to remain below ±0.02 rad (a reasonablebias stability requirement), the temperature could desirably becontrolled to about ±0.003° C. This can be a significant engineeringtask, which increases the complexity, power consumption, and cost of theultimate sensor system.

This same approach has also been used in other ways, for example byplacing the FBG inside a laser cavity, as described in K. P. Koo and A.D. Kersey, “Bragg grating-based laser sensors systems withinterferometric interrogation and wavelength division multiplexing,” J.Lightwave Technol., Vol. 13, Issue 7 (July 1995), to increase thedependence of the wavelength shift on the perturbation applied to theFBG. However, the difficulty arising from the desire to stabilize thetemperature of the imbalanced MZ interferometer remains the same. Tosummarize, a greater discrimination in variations of λ_(Bragg) can beactuated by increasing the length mismatch, but this comes at the priceof a greater instability in the MZ interferometer.

Certain embodiments described herein advantageously utilize new modes ofoperation of an FBG sensor. These new modes provide several substantialbenefits over the previous utilization of FBGs as sensors in theBragg-reflection mode, the largest of which being a greatly increasedsensitivity to a measurand (example, a strain) for a given FBG length,and/or a greatly reduced FBG length for a given sensitivity. In certainembodiments, the sensitivity increase and/or the length reduction are inthe range of a factor of 1 to several orders of magnitude.

Two example optical devices 10 in accordance with certain embodimentsdescribed herein are shown schematically in FIGS. 6A and 6B. In each ofFIGS. 6A and 6B, an optical device 10 comprises an FBG 20 comprising asubstantially periodic refractive index modulation along the length ofthe FBG 20. The FBG 20 has a power transmission spectrum comprising aplurality of local transmission minima. Each pair of neighboring localtransmission minima has a local transmission maximum therebetween. Thelocal transmission maximum has a maximum power at a transmission peakwavelength. The optical device 10 comprises a narrowband optical source30 in optical communication with a first optical path 31 and a secondoptical path 32. The narrowband optical source 30 is configured togenerate light having a wavelength between two neighboring localtransmission minima. The wavelength is at or in the vicinity of a localtransmission maximum, or is at or in the vicinity of a wavelength atwhich the power transmission spectrum has a maximum slope between alocal transmission maximum and either one of the two local transmissionminima neighboring the local transmission maximum.

As used herein, the term “at or in the vicinity of” with regard to aparticular wavelength has its broadest reasonable interpretation,including but not limited to, at the particular wavelength or at awavelength sufficiently close to the particular wavelength such that theperformance of the optical device 10 is substantially equivalent to theperformance of the optical device 10 at the particular wavelength. Forexample, for a wavelength to be “at or in the vicinity of” a particularwavelength can mean that the wavelength is within quantity Δ of theparticular target wavelength, where Δ is a fraction of the FWHMlinewidth of the transmission peak. This fraction can be, for example1%, or 5%, or 10%, or 20%, depending on the application requirement. Forexample, for Δ=10%, if the FWHM linewidth is 2 pm, a wavelength within0.2 pm of a particular target wavelength is considered to be in thevicinity of this target wavelength, and a wavelength that is 2 pm awayfrom this target wavelength is not considered to be in the vicinity ofthis target wavelength.

In certain embodiments, the optical device 10 is an optical sensor andfurther comprises at least one optical detector 40 in opticalcommunication with the FBG 20. The light generated by the narrowbandoptical source 30 is split into a first portion 33 a and a secondportion 33 b. The first portion 33 a is transmitted along the firstoptical path 31 extending along and through the length of the FBG 20. Incertain embodiments, the at least one optical detector 40 is configuredto receive the first portion 33 a, the second portion 33 b, or both thefirst and second portions 33 a, 33 b.

In certain embodiments, the wavelength of the light generated by thenarrowband optical source 30 is at or in the vicinity of a transmissionpeak wavelength of a local transmission maximum such that the FBG 20transmits a substantial fraction of the incident light from thenarrowband optical source 30. In certain such embodiments, asschematically illustrated by FIG. 6A, the first portion 33 a compriseslight incident on the FBG 20 from the narrowband optical source 30 andtransmitted along the FBG 20, and the second portion 33 b compriseslight which does not substantially interact with the FBG 20. The firstportion 33 a therefore is substantially affected by perturbationsapplied to the FBG 20 while the first portion 33 a is transmitted alongthe FBG 20, while the second portion 33 b is substantially unaffected bythe perturbations applied to the FBG 20.

In certain other embodiments, the wavelength of the light generated bythe narrowband optical source 30 is between a local transmission maximumand one of the two neighboring local transmission minima on either sideof the local transmission maximum, such that the FBG 20 transmits asubstantial fraction of the incident light from the narrowband opticalsource 30 and reflects a substantial fraction of the incident light fromthe narrowband optical source 30. In certain such embodiments, asschematically illustrated by FIG. 6B, the first portion 33 a compriseslight incident on the FBG 20 from the narrowband optical source 30 andtransmitted along the FBG 20, and the second portion 33 b compriseslight which is reflected from the FBG 20. In certain embodiments, thefirst portion 33 a therefore is substantially affected by perturbationsapplied to the FBG 20 while the first portion 33 a is transmitted alongthe FBG 20, and the second portion 33 b is substantially affected by theperturbations applied to the FBG 20 while the second portion 33 b isreflected from the FBG 20.

As described more fully below, the light generated by the narrowbandoptical source 30 is selected to be at a wavelength at which the lighttransmitted along the FBG 20 has a slower group velocity than does lightat most other wavelengths propagating through the FBG 20. For example,in certain embodiments, the wavelength of the light generated by thenarrowband optical source 30 can be selected such that the ratio of thespeed of light in vacuum (about 3×10⁵ km/s) to the group velocity of thelight transmitted through the FBG 20 is greater than 5, greater than 10,greater than 30, greater than 50, greater than 100, greater than 300,greater than 500, greater than 1,000, greater than 3,000, greater than5,000, greater than 10,000, greater than 30,000, greater than 50,000,greater than 100,000, greater than 300,000, greater than 500,000, orgreater than 1,000,000. In certain other embodiments, the wavelength ofthe light generated by the narrowband optical source 30 can be selectedsuch that the ratio of the speed of light in vacuum (about 3×10⁵ km/s)to the group velocity of the light transmitted through the FBG 20 isbetween 5 and 10, between 5 and 30, between 10 and 50, between 30 and100, between 50 and 300, between 100 and 500, between 300 and 1,000,between 500 and 3,000, between 1,000 and 5,000, between 3,000 and10,000, between 5,000 and 30,000, between 10,000 and 50,000, between30,000 and 100,000, between 50,000 and 300,000, between 100,000 and500,000, between 300,000 and 1,000,000, between 500,000 and 3,000,000,or between 1,000,000 and 5,000,000.

In certain embodiments, the substantially periodic refractive indexmodulation in the FBG 20 has a constant period along the length of theFBG 20. In certain other embodiments, the substantially periodicrefractive index modulation has a period that varies along the length ofthe FBG 20, as in chirped gratings. In some embodiments, the amplitudeof the index modulation can vary along the length, as in apodizedgratings.

The FBG 20 can be fabricated by exposing the core of an optical fiber toa spatially modulated UV beam, or by many other means. The indexmodulation can be sinusoidal, or take any number of other spatialdistributions. In certain embodiments, the optical fiber is aconventional single-mode fiber such as the SMF-28® optical fiberavailable from Corning, Inc. of Corning, N.Y. However, the fiber inother embodiments is a multimode fiber. In certain other embodiments,the fiber is doped with special elements to make it substantiallyphotosensitive (e.g., substantially responsive to UV light) such thatexposure to a spatially varying light induces a desired modulation inthe refractive index. The fiber can be made of silica, hydrogen-loadedsilica, phosphate glass, chalcogenide glasses, or other materials.

The index perturbation or modulation of the grating in the FBG 20 can beweak (e.g., Δn≈10⁻⁵) or very high (e.g., Δn≈0.015). The index grating ofthe FBG 20 is usually confined to the core, although in some cases italso extends into the cladding immediately surrounding the core. The FBG20 is typically a few mm to a few cm in length, although the FBG 20 inexcess of 1 meter in length or as short as 1 mm have been made.

In certain embodiments, the narrowband optical source 30 comprises asemiconductor laser, e.g., Er—Yb-doped fiber lasers with a wavelengthrange between 1530 nm-1565 nm from NP Photonics in Tucson, Ariz. Inother embodiments, the narrowband optical source 30 comprises a Nd:YAGlaser with a wavelength at 1064.2 nm. In certain embodiments, thenarrowband optical source 30 has a linewidth less than or equal to 10⁻¹³meters. Other wavelengths (e.g., 1.3 microns) and other linewidths arealso compatible with certain embodiments described herein.

In certain embodiments, the light generated by the narrowband opticalsource 30 is split into a first portion 33 a and a second portion 33 b.The first portion 33 a is transmitted along the first optical path 31extending along the length of the FBG 20. The second portion 33 b istransmitted along the second optical path 32 not extending along thelength of the FBG 20. In certain embodiments, as shown in FIG. 6A, thefirst optical path 31 is different from the second optical path 32. Forexample, as shown in FIG. 6A, the first optical path 31 does not overlapthe second optical path 32. For certain other embodiments, the firstoptical path 31 and the second optical path 32 may overlap one another.For example, as shown in FIG. 6B, the first optical path 31 and thesecond optical path 32 both include a common portion between thenarrowband optical source 30 and the FBG 20. In certain embodiments, thefirst optical path 31 and/or second optical path 32 may transverse freespace or various optical elements. For example, one or both of the firstoptical path 31 and the second optical path 32 could transverse anoptical element, e.g., a fiber coupler as described more fully below. Incertain embodiments, the first optical path 31 and/or the second opticalpath 32 may transverse regions with different refractive indices. Forexample, in FIG. 6A, the first optical path 31 transverses the FBG 20having a substantially periodic refractive index modulation along thelength of the FBG 20.

In certain embodiments, the optical device 10 comprises at least oneoptical detector 40 in optical communication with the FBG 20. The atleast one optical detector 40 is configured to receive the first portion33 a of light, the second portion 33 b of light, or both the first andsecond portions 33 a, 33 b of light. In certain embodiments, the opticaldetector 40 is a New Focus general purpose photodetector Model 1811,low-noise photodetector. However, the optical detector 40 may be one ofa variety of low-noise photodetectors well known in the art, althoughdetectors yet to be devised may be used as well.

In certain embodiments, a mode of operation, referred to herein as theslow-light transmission mode, can be used (e.g., with the structureschematically illustrated by FIGS. 6A and 7). In these embodiments,light with a narrowband spectrum is launched into the FBG 20, with awavelength at (e.g., on or near) a wavelength where the FBG 20 mostlytransmits, rather than mostly reflects, light. For example, thewavelength of the light is selected to be at (e.g., on or near) atransmission peak wavelength corresponding to a local transmissionmaximum of the power transmission spectrum of the FBG 20. The possiblelocations of these two wavelengths, referred to herein as λ₁ and λ′₁, isdiscussed in greater detail below, in particular in relation to FIGS.9A-9H. At these wavelengths, light experiences a significant groupdelay, i.e., it travels with a much slower group velocity than the groupvelocity of light at wavelengths further away from the bandgap of theFBG 20. For example, slow-light group velocities can be as low as 300km/s, while non-slow-light group velocities are typically around 207,000km/s for light traveling in silica. Light with this slower groupvelocity in the vicinity of the edges of the bandgap of the FBG 20 isreferred to herein as slow light. Slow light has been investigatedpreviously in other contexts, in particular, to evaluate the potentialuse of an FBG for dispersion compensation, for example in opticalcommunication systems. See, e.g., F. Ouellette, P. A. Krug, T. Stephens,G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensationusing chirped sample fibre Bragg gratings,” Electronic Letters, Vol. 31,No. 11 (May 1995).

In certain embodiments, a benefit of the slow-light transmission mode ofoperation is that in the vicinity of a slow-light wavelength, e.g., λ₁or λ′₁, the power transmission has a local maximum (e.g., it can beclose to or equal to 1). Consequently, the loss experienced by thesignal as it propagates along or through the FBG 20 is small. In certainembodiments, another benefit is that at or in the vicinity of either oneof the slow-light wavelengths λ₁ and λ′₁, a perturbation (e.g., astrain) applied to the FBG 20 on light traveling through the FBG 20modifies the phase of the light traveling through the FBG 20, not itsamplitude. To be more exact, in certain embodiments, the perturbationmodifies to first order the phase of light, and to second order theamplitude of the light. This is in contrast to the Bragg-reflection modeof an FBG, in which the perturbation to the FBG modifies the frequencyof the light that is maximally reflected. Consequently, in certainembodiments using the slow-light transmission mode (e.g., FIGS. 6A and7), the FBG 20 can serve as a phase sensor, e.g., it can be placeddirectly in one of any number of interferometers to convert the phasemodulation induced by the perturbation (the measurand) into a powerchange (which is the quantity the user measures).

FIG. 7 schematically illustrates an example configuration comprising anominally balanced MZ interferometer. The configuration of FIG. 7 is anexample of the general configuration schematically shown in FIG. 6A. InFIG. 7, the optical device or sensor 10 comprises a first fiber coupler51 in optical communication with the narrowband light source 30, thefirst optical path 31, and a second optical path 32 not extending alongand through the FBG 20. The light generated by the narrowband opticalsource 30 is split by the first fiber coupler 51, e.g., with a 3-dBpower-splitting ratio, into the first portion 33 a and the secondportion 33 b. In this embodiment, the first portion 33 a is transmittedalong the first optical path 31, and the second portion 33 b istransmitted along the second optical path 32. The first portion 33 apropagates along the FBG 20 while the second portion 33 b does notsubstantially interact with the FBG 20. In this embodiment, the firstportion 33 a includes information regarding the perturbation of the FBG20, while the second portion 33 b remains unaffected by suchperturbation.

In FIG. 7, the optical sensor 10 further comprises a second fibercoupler 52, e.g, with a 3-dB power-splitting ratio, in opticalcommunication with the first optical path 31 and the second optical path32. The first portion 33 a and the second portion 33 b are recombined bythe second fiber coupler 52 and transmitted to the at least one opticaldetector 40. This recombination allows the first portion 33 a and thesecond portion 33 b to interfere with one another, producing a combinedsignal that contains information regarding the phase difference betweenthe first portion 33 a and the second portion 33 b. In certainembodiments, the at least one optical detector 40 comprises a singleoptical detector at one of the output ports of the second fiber coupler52. In certain other embodiments, as schematically illustrated by FIG.7, the at least one optical detector 40 comprises a first opticaldetector 40 a at one output port of the second fiber coupler 52 and asecond optical detector 40 b at the other output port of the secondfiber coupler 52. The signals detected by these two optical detectors 40a, 40 b vary in opposite directions from one another (e.g., when thedetected power at one output port of the second fiber coupler 52increases, the detected power at the other output port of the secondfiber coupler 52 decreases) and the difference between the outputs fromthe two optical detectors 40 a, 40 b can be used as the sensor signal.Such a detection scheme can provide various advantages, including commonmode rejection and a higher signal. In certain embodiments, the phasedifference is indicative of an amount of strain applied to the FBG 20.In certain other embodiments, the phase difference is indicative of atemperature of the FBG 20.

In certain embodiments, using a balanced MZ interferometer configurationwith slow light, as schematically illustrated by FIG. 7, allows precisedetection and measurement of the perturbation applied to the FBG bydetecting and measuring the phase difference between the first portion33 a and the second portion 33 b. In contrast, in the Bragg-reflectionmode, a wavelength (or frequency) change is detected (e.g., as shown inFIG. 1) or is converted into a power change, which, for high precision,can be done by stabilizing an imbalanced interferometer (e.g., as shownin FIG. 2). Also, in certain embodiments, use of a balanced MZinterferometer in the slow-light transmission mode advantageously avoidsthe high sensitivity to temperature of an imbalanced MZ interferometer,resulting in a great improvement in the temperature stability of the MZinterferometer, and therefore of its phase bias. This also simplifiesengineering considerably by reducing the amount of temperature controlto be used.

When light travels through a medium and the group velocity is low, thematter-field interaction is increased. Since it takes a longer time forthe light to travel through the medium, the compression of the localenergy density gives rise to enhanced physical effects, including phaseshift. The induced phase dependence on dk shift is significantlyenhanced when the group velocity ν_(g)=dω/dk is small. As shown in M.Solja{hacek over (c)}ić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen,and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement ofnonlinear phase sensitivity,” JOSA B, Vol. 19, Issue 9 (September 2002),this effect can be quantified by relating the phase shift to the groupvelocity:

δφ=L×δk≈L*δω(dω/dk)  (6)

This relationship states that the phase shift is inversely proportionalto the group velocity ν_(g)=dω/dk, or is proportional to the group indexn_(g)=c/ν_(g), where c is the speed of light in vacuum. The main benefitof operating in this slow-light transmission mode in accordance withcertain embodiments described herein, as stated without demonstrationearlier, is that everything else being the same, a given perturbationwill induce a much larger phase perturbation in a device in which lighthas a low group velocity than in a device in which light has a highgroup velocity. As demonstrated below with numerical simulations, anoptical sensor 10 comprising an FBG 20 operated in the slow-lighttransmission mode in accordance with certain embodiments describedherein can therefore exhibit a far greater sensitivity to any measurandthat alters the phase of a signal traveling in the grating.

The Mach-Zehnder (MZ) interferometer in the configuration of FIG. 7 isonly one of many interferometers that can be used to convert the phaseshift induced by the perturbation on light traveling along the FBG 20into an intensity change. Any interferometer that converts a phasemodulation into an amplitude modulation can be used instead of a MZinterferometer. For example the interferometer can be a Michelsoninterferometer, a Fabry-Perot interferometer, or a Sagnac interferometer(if the perturbation is time-dependent). For example, in certainembodiments, the optical sensor 10 comprises a fiber loop (e.g., aSagnac interferometric loop) comprising the fiber Bragg grating which islocated asymmetrically. The first optical path extends in a firstdirection along the fiber loop, and the second optical path extends in asecond direction along the fiber loop, the second direction opposite tothe first direction. In certain such embodiments, the optical sensor 10further comprises at least one fiber coupler optically coupled to thenarrowband optical source and the fiber loop, wherein the lightgenerated by the narrowband optical source is split by the at least onefiber coupler into the first portion and the second portion such thatthe first portion propagates along the first optical path and the secondportion propagates along the second optical path. The first portion andthe second portion are recombined by the at least one fiber couplerafter propagating along and through the length of the fiber Bragggrating. The at least one optical detector comprises an optical phasedetector configured to receive the recombined first and second portionsand to detect the phase difference between the first portion and thesecond portion. Such a Sagnac configuration can be used to detecttime-varying perturbations to the FBG 20. In certain embodiments, the atleast one fiber coupler is configured to allow detection of the power atthe reciprocal output port.

FIGS. 6B and 8 schematically illustrate a second mode of operationdisclosed and referred to herein as the slow-light reflection mode. Theconfiguration of FIG. 8 is an example of the general configurationschematically shown in FIG. 6B. The optical device 10 comprises an FBG20. The FBG 20 comprises a substantially periodic refractive indexmodulation along a length of the FBG 20. The FBG 20 has a powertransmission spectrum comprising a plurality of local transmissionminima. Each pair of neighboring local transmission minima has a localtransmission maximum therebetween. The local transmission maximum has amaximum power at a transmission peak wavelength. The optical sensor 10comprises a narrowband optical source 30 in optical communication with afirst optical path 31 and a second optical path 32. The narrowbandoptical source 30 is configured to generate light having a wavelengthbetween two neighboring local transmission minima (e.g., between a localtransmission maximum and a local transmission minimum next to the localtransmission maximum). In certain embodiments, the optical device 10 isan optical sensor further comprising at least one optical detector 40 a,40 b in optical communication with the FBG 20. The light generated bythe narrowband optical source 30 is split by the FBG 20 into a firstportion 33 a and a second portion 33 b. The first portion 33 a istransmitted along the first optical path 31 extending along the lengthof the FBG 20. The second portion 33 b is transmitted along the secondoptical path 32 and reflected from the FBG 20 and thus not extendingalong the length of the FBG 20, as schematically illustrated by FIG. 8.In certain embodiments, at least one optical detector 40 a 40 b isconfigured to receive the first portion 33 a, the second portion 33 b,or both the first and second portions 33 a, 33 b.

In the embodiment shown in FIG. 8, the light generated by the narrowbandoptical source 30 transverses a fiber coupler 51. However, unlike theembodiment shown in FIG. 7, the first optical path 31 and the secondoptical path 32 overlap one another. The first optical path 31 of FIG. 8extends from the narrowband optical source 30, through the fiber coupler51, and along and through the length of the FBG 20. The second opticalpath 32 of FIG. 8 extends from the narrowband optical source 30, throughthe fiber coupler 51, to the FBG 20 where the light is reflected backtowards the fiber coupler 51. The transmitted first portion 33 a isformed by light incident on the FBG 20 in a first direction and uponinteracting with the FBG 20, some of this incident light constructivelyinterferes forward (in the first direction). The reflected secondportion 33 b is formed by light incident on the FBG 20 in a firstdirection and upon interacting with the FBG 20, some of this incidentlight constructively interferes backwards (in a second directionopposite to the first direction). In this embodiment, the second portion33 b does not propagate along the FBG 20 because it is reflected fromthe FBG 20. The optical detector 40 b is configured to receive the firstportion 33 a, while the optical detector 40 a is configured to receivethe second portion 33 b after the second portion 33 b transverses backthrough the fiber coupler 51.

In certain embodiments, the FBG 20 is interrogated with a narrowbandlaser 30 and the first portion 33 a is transmitted along the FBG 20 andthe second portion 33 b is reflected from the FBG 20. The wavelength ofthe light interrogating the FBG 20 is selected to be between a localtransmission maximum of the power transmission spectrum (e.g., λ₁, λ₂,λ₃, λ′₁, λ′₂, λ′₃, or λ_(i) or λ′_(i) with i≧1, referring to FIGS. 9Aand 9E, discussed more fully below) and a local transmission minimumnext to the local transmission maximum. In certain such embodiments, thelight incident on the FBG 20 has a wavelength λ_(A) and λ′_(A) at or inthe vicinity of the steepest portion of the FBG 20 reflection peak(e.g., λ_(a), λ_(b), λ_(c), λ_(d), λ_(e), λ_(f), λ′_(a), λ′_(b), λ′_(c),λ′_(d), λ′_(e), λ′_(f), or other such wavelengths between thewavelengths λ′_(i) with i≧1, referring to FIGS. 9A and 9E, discussedmore fully below).

For example, in certain embodiments, the FBG 20 reflects light in arange of wavelengths encompassing the Bragg wavelength from a first edgewavelength (e.g., the transmission peak wavelength λ₁ of a first localtransmission maximum, discussed more fully below) to a second edgewavelength (e.g., the transmission peak wavelength λ′₁ of a second localtransmission maximum, discussed more fully below). The reflected lighthas a maximum intensity at a reflection peak wavelength (e.g., the Braggwavelength) within the bandgap (e.g., between the first edge wavelengthand the second edge wavelength). The region between the two transmissionpeak wavelengths λ₁ and λ′₁ can be considered to be a local transmissionminimum of the power transmission spectrum of the FBG 20. In certainsuch embodiments, the wavelengths can be selected to be on the edge ofthe resonance or slow-light peaks at which the power transmission is aselected fraction (e.g., about one-half, or in a range between ⅕ and ⅘)of the maximum value of the power transmission at the transmission peakwavelengths λ₁ and λ′₁ of the first or second local transmission maxima.

When an external perturbation is applied to the FBG 20, the reflectionpeak shifts in wavelength. This shift of λ_(Bragg) results in a changein the first portion 33 a transmitted by the FBG 20 and in the secondportion 33 b reflected by the FBG 20, for example, in the power of thereflected light at the wavelength of the light incident on the FBG 20.In certain embodiments, the at least one optical detector 40 comprises aphotodiode 40 a configured to receive and to detect the optical power ofthe second portion 33 b. As shown in FIG. 8, the second portion 33 b ofthe laser signal reflected by the FBG 20 is separated from the FBG'sinput port by a fiber coupler 51 (e.g., a fiber coupler with about a3-dB power-splitting ratio), and the optical power of the second portion33 b is measured by a photodetector 40 a. In certain embodiments, the atleast one optical detector 40 comprises a photodiode 40 b configured toreceive and to detect the optical power of the first portion 33 a. InFIG. 8, the change in power can be detected at the output of the FBG 20with a photodetector 40 b in optical communication with the output ofthe FBG 20.

In certain embodiments, the detected optical power is indicative of anamount of strain applied to the FBG 20. In certain other embodiments,the detected optical power is indicative of a temperature of the FBG 20.

In certain embodiments operating in a slow-light reflection mode, thesignal experiences a slow group velocity as it travels through the FBG20, although not quite as slow as certain embodiments in the slow-lighttransmission mode of FIG. 7. Certain such embodiments therefore alsoadvantageously provide a greatly increased sensitivity over an FBGoperated in the Bragg-reflection mode. In addition, because theslow-light reflection mode does not involve measuring a wavelengthshift, unlike the Bragg-reflection mode (e.g., FIG. 2), it does notutilize an imbalanced MZ interferometer, thereby eliminating the issueof MZ interferometer thermal stability and simplifying the designsignificantly. In the slow-light reflection mode of operation, inaccordance with certain embodiments described herein, the powertransmission of the FBG 20 is not as high as in certain embodiments inthe slow-light transmission mode. However, the transmission is stillhigh (˜70%, excluding losses, depending on details of the design), sothat the loss experienced by the signal as it propagates through the FBG20 is still small. Therefore, in certain embodiments, the optical powertransmitted by the FBG 20 can be detected and measured (e.g., by thephotodiode 40 b) to measure the perturbation applied to the FBG 20.

The sensitivity of certain embodiments of an optical sensor 10 operatedin one of the new reflection and transmission modes described hereindepends directly on how slow the group velocity of the light can be madein the FBG 20. A number of computer simulations described belowillustrate this principle and quantify the magnitude of the sensitivityimprovement brought about by certain embodiments of these new modes ofoperation. For comparison, these simulations also model the sensitivityof an FBG, in the Bragg-reflection mode outlined above to a particularmeasurand, namely temperature. The results would have been substantiallythe same had the simulation modeled the effect of another measurand,such as a strain. These simulations utilized well-known expressions forthe phase of a signal traveling through a grating of known parameters(see, e.g., A. Yariv and P. Yeh, Optical waves in crystals: propagationand control of laser radiation, pp. 155-214 (New York: Wiley 1984)),namely a sinusoidal index modulation with a period Λ and an amplitudeΔn, a grating length L, and a uniform, small temperature change ΔT.

FIGS. 9A-9D show the calculated properties of a temperature sensor usinga sinusoidal FBG with the following parameter values: Δn=1.5×10⁻⁴, L=1cm, and Λ=0.37 μm (which gives a Bragg wavelength λ_(Bragg) around 1.064μm), calculated assuming a lossless grating. FIGS. 9E-9H show thecalculated properties of a temperature sensor with a sinusoidal FBG withΔn=2.0×10⁻⁴, L=2 cm, loss=1 m⁻¹, and Λ=0.53 μm (which gives a Braggwavelength λ_(Bragg) around 1.55 μm), calculated assuming a losslessgrating. FIGS. 9A and 9E illustrate the power transmission of thisgrating in the vicinity of λ_(Bragg), calculated at two temperatures,namely room temperature (300K) (denoted in FIGS. 9A-9C and 9E-9G asunheated). For the Bragg wavelength of 1.064 microns shown in FIGS.9A-9C, the heated curve corresponds to room temperature plus ΔT=0.01° C.For the Bragg wavelength of 1.55 microns shown in FIGS. 9A-9C, theheated curve corresponds to room temperature plus ΔT=0.01° C. Becausethe temperature change is very small, the heated and unheated curves ofFIGS. 9A-9C and 9E-9G are so close to each other that they cannot bedistinguished on the graphs. FIGS. 9A-9C and 9E-9G illustrate thewavelength dependence of the FBG power transmission, phase, and groupindex for the Bragg wavelengths of 1.064 microns and 1.55 microns,respectively. In the vicinity of λ_(Bragg), the FBG acts as a reflector,and its transmission is expectedly close to zero, as shown by FIGS. 9Aand 9E. This low transmittance region, where the reflectivity is high,constitutes approximately the bandgap of the FBG. The full width at halfmaximum (FWHM) of the bandgap of this grating for λ_(Bragg)=1.064 μm isapproximately 126 pm, as shown by FIG. 9A, and the FWHM of the bandgapfor λ_(Bragg)=1.55 μm is approximately 202 pm, as shown by FIG. 9E.Outside this bandgap region, the transmission reaches a first resonancepeak, then it is oscillatory, with diminishing amplitudes, further awayfrom λ_(Bragg). Far enough from λ_(Bragg) (outside the range shown inthe figures), the transmission goes asymptotically to near unity.

As mentioned earlier, the first wavelength where the transmissionreaches short a resonance peak is referred to herein as λ₁ (on the shortwavelength side of λ_(Bragg)) and λ′₁ (on the long wavelength sideλ_(Bragg)). The higher order wavelengths where the transmission reachesa resonance peak are referred to as λ_(i) (on the short wavelength sideof λ_(Bragg)) and λ′_(i) (on the long wavelength side λ_(Bragg)), wherei≧2. In certain embodiments, the narrowband optical source generateslight having a wavelength at or in the vicinity of one of the localtransmission maxima (e.g., also referred to herein as resonance peaks orslow-light peaks, and which can be denoted by λ₁, λ₂, λ₃, λ₄, etc., andλ′₁, λ′₂, λ′₃, λ′₄, etc.). In certain embodiments, the narrowbandoptical source generates light having a wavelength (denoted by λ_(a),λ_(b), λ_(d), etc., and λ′_(a), λ′_(b), λ′_(c), λ′_(d), etc.) betweenone of the local transmission maxima (e.g., also referred to herein asresonance peaks or slow-light peaks, and which can be denoted by λ₁, λ₂,λ₃, λ₄, etc., and λ′₁, λ′₂, λ′₃, λ′₄, etc.) and a neighboring localtransmission minimum.

For example, in certain embodiments in which the power transmissionspectrum has a first local transmission maximum λ₁ between a first localtransmission minimum comprising the Bragg wavelength and a second localtransmission minimum on a short wavelength side of the Bragg wavelength,and a second local transmission maximum λ₂ between the second localtransmission minimum and a third local transmission minimum on the shortwavelength side of the Bragg wavelength, the wavelength of the lightgenerated by the narrowband optical source can be selected to be betweenthe first local transmission minimum and the second local transmissionminimum, at the first local transmission maximum, between the firstlocal transmission maximum λ₁ and either the first local transmissionminimum or the second local transmission minimum, between the secondlocal transmission minimum and the third local transmission minimum, atthe second local transmission maximum, or between the second localtransmission maximum and either the second local transmission minimum orthe third local transmission minimum. Similarly, the wavelength can beselected to be on the short wavelength side of the Bragg wavelength atthe third local transmission maximum, the fourth local transmissionmaximum, or between either the third or fourth local transmissionmaximum and a neighboring local transmission minimum.

As another example, in certain embodiments in which the powertransmission spectrum has a first local transmission maximum λ′₁ betweena first local transmission minimum comprising the Bragg wavelength and asecond local transmission minimum on a long wavelength side of the Braggwavelength, and a second local transmission maximum λ′₂ between thesecond local transmission minimum and a third local transmission minimumon the long wavelength side of the Bragg wavelength, the wavelength ofthe light generated by the narrowband optical source can be selected tobe between the first local transmission minimum and the second localtransmission minimum, at the first local transmission maximum, betweenthe first local transmission maximum and either the first localtransmission minimum or the second local transmission minimum, betweenthe second local transmission minimum and the third local transmissionminimum, at the second local transmission maximum, or between the secondlocal transmission maximum and either the second local transmissionminimum or the third local transmission minimum. Similarly, thewavelength can be selected to be on the long wavelength side of theBragg wavelength at the third local transmission maximum, the fourthlocal transmission maximum, or between either the third or fourth localtransmission maximum and a neighboring local transmission minimum.

FIGS. 9B and 9F illustrate the calculated phase of a narrowband signalafter it has traveled through this grating, as a function of wavelength.This calculation was again conducted for the temperatures used for FIGS.9A and 9E. Around the particular wavelengths λ₁ and λ′₁, and λ₂ and λ′₂,the phase varies more rapidly with wavelength than around the center ofthe curve (around λ_(Bragg), where the FBG reflects strongly). Thisincreased dependence of phase on wavelength at or near the edgewavelength is the result of a larger group delay of the signal as ittravels through the grating. In other words, in the vicinity, e.g., ±5pm, of these two wavelengths the grating supports slow light.

FIGS. 9C and 9G plot the group index of light traveling through this FBGcalculated as a function of wavelength by applying Equation 6 to thephase dependence on wavelength of FIGS. 9B and 9F. FIGS. 9C and 9Gillustrate that in the vicinity of λ₁ and λ′₁, the group index n_(g)increases markedly. The same is true in the vicinity of othertransmission resonances, such as λ₂ and λ′₂ shown in the figures, butalso of other resonances λ_(i) and λ′_(i) outside of the range ofwavelengths shown in the figures. Specifically, in the example gratingwith λ_(Bragg)=1.064 μm, n_(g) reaches a value of about 4.2 at or nearthe edge wavelength. In the fiber, and in the FBG for wavelengths farfrom λ_(Bragg), the group index of light is approximately c/n, or about207,000 km/s (this value depends weakly on optical wavelengths). Incontrast, around the two edge wavelengths, the group index is only abouta factor of 4.2 smaller than the speed of light in free space, or about71,400 km/s. In another example grating with λ_(Bragg)=1.55 μm, n_(g)reaches a value of about 8.7 at or near the edge wavelength or a groupvelocity of about 34,500 km/s.

The power transmission spectrum, transmitted phase, and group index ofan FBG with a sinusoidal index perturbation exhibiting the generalbehavior outlined in FIGS. 9A-9C had been previously reported in adifferent context. See M. Lee et al, “Improved slow-light delayperformance of a broadband stimulated Brillouin scattering system usingfiber Bragg gratings,” Applied Optics Vol. 47, No. 34, pp. 6404-6415,Dec. 1, 2008. In this reference, the authors modeled, through numericalsimulations, an FBG with similar parameters as used in FIGS. 9A-9C,namely a sinusoidal index modulation with a length L=2.67 cm, an indexcontrast Δn of 10⁻⁴, and a grating period Λ=533 nm (a Bragg wavelengthof 1550 nm). Their conclusion was that the light traveling through theFBG exhibits a group delay higher than normal when the frequency of thelight is centered in the vicinity of the first transmission peak on theside of the Bragg wavelength reflection peak. This property was used inthat reference to increase the group delay in an SBS-based optical delayline without increasing the power consumption, by adding one or two FBGsto the SBS delay line. However, this reference fails to recognize orteach that the group delay increases nonlinearly with length. In fact,the reference discloses that the group delay doubles when using twogratings instead of one, in contrast to aspects of certain embodimentsdescribed herein. In particular, as described more fully below, thegroup delay increases as a high power of the grating length. Inaddition, this reference remains silent on various aspects describedherein, including but not limited to: (1) the desirability of increasingthe index contrast in order to increase the group delay, (2) the phaseaccumulating faster at this wavelength, (3) the existence of otherresonant wavelengths (where the grating transmission exhibits a localmaximum) and the possibility of operating at these wavelengths, and (4)the use of an FBG as a sensor in the slow-light mode and its benefits,as well as means of optimizing its performance characteristics.

In certain embodiments described herein, the FBG is designed orconfigured to produce extremely large group delays, or equivalently,extremely large group indices, which results in extremely highsensitivity when this FBG is used as a sensor in one of the slow-lightmodes of operation described herein. In comparison, previous research onFBGs has produced relatively small group indices. For example, in M. Leeet al, previously cited, the maximum group index calculated from FIG. 2(a) in that reference is about 3.3. As another example, in Joe T. Mok etal, “Dispersionless slow light using gap solitons,” Nature Physics, Vol.21, pp. 775-780, November 2006, a group index of about 5 is reported inan apodized FBG of 10 cm length with an index contrast Δn=1.53×10⁻⁴. Incontrast, certain embodiments described herein utilize a revolutionaryconcept that enables the production of FBGs with group indices in therange of several hundred, if not much higher, as described more fullybelow.

Certain embodiments described herein advantageously provide FBGs withconsiderably larger group index, in the range of 10 s to 100 s, or more.Such gratings can be used for producing fiber sensors with significantlyincreased sensitivity, with improvements of tens to hundreds, or more,compared to existing FBG-based sensors, for most measurands, asdescribed below. They can also be used for any application utilizing orbenefiting from a large group index, or a large group delay, including,but not limited to, solitons, group delay lines, dispersioncompensation, and optical filters.

Based on Equation 6, and in the light of the group index value of about4.2 that can be achieved with the FBG of FIGS. 9A-9C, or the group indexvalue of about 8.7 that can be achieved with the FBG of FIGS. 9E-9G, thesensitivity of the slow-light FBG sensor in accordance with certainembodiments described herein to temperature is significantly greaterthan when this same FBG is used in the Bragg-transmission mode. FIGS.9A-9C each show two curves for λ_(Bragg)=1.064 μm, each one calculatedat two temperatures spaced by ΔT=0.01° C. and FIGS. 9E-9G each show twocurves for λ_(Bragg)=1.55 μm, each one calculated at two temperaturesspaced by ΔT=0.01° C. By taking the difference between the two phasecurves of FIG. 9B and dividing by ΔT, since ΔT is small, one obtains aclose approximation of the derivative of the phase with respect totemperature dφ/dT for λ_(Bragg)=1.064 μm, as shown in FIG. 9D.Similarly, the difference between the two phase curves of FIG. 9F can bedivided by ΔT to obtain a close approximation of the derivative of thephase with respect to temperature dφ/dT for λ_(Bragg)=1.55 μm, as shownin FIG. 9H. The maximum sensitivity occurs in the vicinity of λ₁ andλ′₁. At either of these wavelengths, the group index is 4.2 and thesensitivity dφ/dT is 2.9 rad/° C. for λ_(Bragg)=1.064 μm, and the groupindex is 8.7 and the sensitivity dφ/dT is 8.1 rad/° C. forλ_(Bragg)=1.55 μm. Hence at these slow-light wavelengths, thesensitivity to temperature is quite large.

FIGS. 10A and 10B illustrate in greater detail the relationship betweenthe phase sensitivity and the transmission for λ_(B)=1.064 μm and forλ_(B)=1.55 μm, respectively, calculated assuming a lossless grating. Itshows that the sensitivity is maximum not exactly at λ₁ and λ′₁ (wherethe power transmission is equal to one), but in their vicinity. Sincethe transmission is by definition maximum (and equal to unity) at thesetwo wavelengths, FIGS. 10A and 10B illustrate that (at least in theseexamples) there is not a single wavelength that maximizes both thetransmission (which is desirable to minimize the loss experienced by thesignal as it travels through the grating) and the group index (which isdesirable to maximize the sensitivity). However, the wavelengths atwhich the transmission and group index are maximized are in the vicinityof one another, so the compromise to be made is relatively small. Forexample, for λ_(Bragg)=1.064 μm, at the first transmission resonancepeak, the transmission is unity, the group index is 4.0, and thesensitivity dφ/dT is 2.6 rad/° C. At the wavelength where the groupindex is maximum (and equal to 4.2) the sensitivity is 2.85 rad/° C. andthe transmission is equal to 94%. As another example, for λ_(Bragg)=1.55μm, at the first resonance peak (transmission ≈89%) the group index is8.38, and the sensitivity dφ/dT is 7.84 rad/° C. At the wavelength wherethe group index is maximum (and equal to 8.7) the sensitivity is 8.1rad/° C., and the transmission is equal to 85%. When loss is considered,the resonance peaks do not reach 100% power transmission. These valuesdiffer from their respective maxima by less than 10%, so in certainembodiments, the wavelength can be selected to maximize either thetransmission or the group index, depending on criteria imposed by thespecific application targeted. Regardless of the exact operating point,both the transmission and the group index (and thus sensitivity) arenear their respective maximum over a comparatively broad range ofwavelengths, which is a useful feature in certain embodiments, on bothcounts. This qualitative conclusion is valid for a very wide range ofFBG parameter values, even for values that produce considerably highergroup indices (for example, 10⁵) than are used in this particularnumerical example.

The figures discussed above were generated by modeling an FBG with agiven index contrast (Δn=1.5×10⁻⁴ for λ_(Bragg)=1.064 μm and Δn=2.0×10⁻⁴for λ_(Bragg)=1.55 μm). As the index contrast is increased, the groupdelay increases further, and according to Eq. 6 the sensitivity to themeasurand also increases. Since the Δn of an FBG can be considerablyhigher than this modeled value, for example when the FBG is fabricatedin a hydrogen-loaded fiber (e.g., Δn of 0.015, see, e.g., P. J. Lemaire,R. M. Atkins, V. Mizrahi, and W. A. Reed, “High pressure H₂ loading as atechnique for achieving ultrahigh UV photosensitivity and thermalsensitivity in GeO₂ doped optical fibres,” Electronic Letters, Vol. 29,No. 13 (June 1993)), a substantial increase in group delay andsensitivity results from increasing Δn. To quantify this improvement,the sensitivity was computed as a function of index contrast for agrating used in the slow-light transmission configuration in accordancewith certain embodiments described herein. The gratings are assumed tohave zero loss in both wavelengths to illustrate dependence of groupindex and sensitivity on index modulation and length only. FIGS. 11A(for λ_(Bragg)=1.064 μm) and 11B (for λ_(Bragg)=1.55 μm) plot (i) thissensitivity of the slow-light transmission configuration at λ₁ orequivalently λ′₁ along with (ii) the sensitivity for a grating used inthe slow-light reflection configuration at λ_(a) or equivalently λ′_(a)in accordance with certain embodiments described herein, and (iii) for agrating used in the Bragg-reflection configuration, calculated assuminga lossless grating. All three curves for each of FIGS. 11A and 11B werecomputed for an exemplary grating length L=2 cm. In order to performthis comparison, the phase sensitivity was converted into a powersensitivity in the slow-light transmission scheme as follows. When theMZ interferometer is phase biased for maximum sensitivity, the outputpower at one of the output powers of the interferometer is given as P=P₀sin(Δφ/2), when P₀ is the total output power (including both ports) andΔφ is the phase difference between the two arms. When a smallperturbation δΔφ is applied to the FBG placed in one of the two arms ofthe MZ interferometer, the output power varies by δP≈P₀δΔφ/2. Hence thepower sensitivity of the sensor is, by definition:

$\begin{matrix}{{\frac{1}{P_{0}}\frac{P}{T}} = {\frac{1}{2}\frac{\varphi}{T}}} & (7)\end{matrix}$

In other words, it is equal to half the phase sensitivity used above asthe metric for sensitivity.

In the slow-light transmission configuration, for a given gratinglength, below a certain index contrast, the sensitivity is constant.When the index contrast is large enough (typically above about 10⁻⁴),the sensitivity increases as a higher power of Δn. For example, for agrating length of 2 cm operating at 1.064 μm, the power sensitivity totemperature scales as Δn^(1.95). As another example, for a gratinglength of 2 cm operating at 1.55 μm, the power sensitivity totemperature scales as Δn^(1.99). In comparison, in the slow-lightreflection configuration, the sensitivity grows monotonically as theindex contrast is increased (see FIGS. 11A and 11B). For an indexcontrast above about 10⁻⁴ in this example, the sensitivity of the twoslow-light schemes are extremely close to each other.

In contrast, FIGS. 11A (1.064 μm) and 11B (1.55 μm) also show that inthe Bragg-reflection mode, below a certain index contrast (about 10⁻⁵for this particular grating length) the sensitivity is constant, just asin the case of the slow-light transmission scheme. Above this indexcontrast, the sensitivity decreases. The reason for this decrease wasdiscussed above. In these simulations, the sensitivity of theBragg-reflection configuration was maximized by maximizing the lengthimbalance ΔL of the MZ interferometer (e.g., by making the lengthimbalance equal to the coherence length of the light reflected by theFBG). Since this coherence length depends on the index contrast of theFBG (see Equations 2 and 3), this value was adjusted for each value ofΔn used in the simulations. As the index contrast increases, thecoherence length of the reflected light decreases, and the lengthimbalance ΔL decreases, therefore the sensitivity decreases (seeEquation 5).

FIG. 11A shows that for Δn of 1.5×10⁻², which again is attainable inpractice, (see, e.g. Lemaire et al.) the power sensitivity in theslow-light transmission and slow-light reflection schemes forλ_(Bragg)=1.064 μm is as high as ˜6.5×10⁴° C.⁻¹ (upper end of thecurve), corresponding to a phase sensitivity of 1.3×10⁵ rad/° C. This isnearly 110,000 times higher than the best value predicted for a grating(L=2 cm, Δn=10⁻⁵) operated in the Bragg-reflection mode (1.18 rad/° C.).FIG. 11B shows that for Δn of 1.5×10⁻², the power sensitivity in theslow-light transmission and slow-light reflection schemes forλ_(Bragg)=1.55 μm is as high as ˜1.9×10⁴° C.⁻¹, corresponding to a phasesensitivity of 3.8×10⁴ rad/° C. This is ˜46,000 times higher than thebest value predicted value for a grating (L=2 cm, Δn=10⁻⁵) operated inthe Bragg-reflection mode (0.82 rad/° C.).

The reason why the two slow-light configurations exhibit almost the samesensitivity for large Δn (see FIGS. 11A and 11B) is that the group indexof light used at wavelength λ₁ (or λ′₁) in the slow-light transmissionconfiguration and light used at λ₂ (or λ′₂) in the slow-light reflectionconfiguration are almost the same. This can be seen in FIGS. 12A (1.064μm) and 12B (1.55 μm), which plot the group index calculated as afunction of Δn for an FBG of length L=2 cm, calculated assuming alossless grating. These plots were again computed at λ₁ (or equivalentlyλ′₁) for the transmission mode, and λ_(a) (or equivalently λ′_(a)) forthe reflection mode. The slow-light reflection and slow-lighttransmission configurations produce almost the same group index, thelatter being only slightly smaller for the reflection configuration. Inboth schemes, the group index increases with Δn approximately asΔn^(1.95). In the slow-light reflection scheme, the signal wavelength ismore strongly detuned from the wavelength that produces the slowestlight than it is in the slow-light transmission scheme. FIG. 12A alsodemonstrates that it is possible to achieve extremely slow light in anoptical fiber grating with λ_(Bragg)=1.064 μm. In this example, themaximum practical n_(g), which occurs for an FBG with a Δn of 0.015(hydrogen-loaded FBG), is around 10⁵. This corresponds to a groupvelocity of only 3,000 m/s. It is about 20,000 times slower thanpreviously demonstrated in an FBG, experimentally or throughsimulations. By increasing the FBG length from 2 cm (the value used inthis simulation) to 10 cm, this group-index figure is increased byapproximately 5^(1.99)≈25, to 2.5 million—a group velocity of 120 m/s.In the example shown in FIG. 12B for λ_(Bragg)=1.55 μm, again for bothschemes the group index increases with Δn approximately as Δn^(1.99).The maximum practical n_(g), which occurs for an FBG with a Δn of 0.015(hydrogen-loaded FBG), is around 4.2×10⁴. This corresponds to a groupvelocity of only 7,100 m/s. It is about 8,000 times slower thanpreviously demonstrated in an FBG, experimentally or throughsimulations. By increasing the FBG length from 2 cm to 10 cm, thisfigure is increased to approximately one million—a group velocity of 300m/s. This property has tremendous implications for a large number ofapplications, including all of the aforementioned applications (e.g.,including but not limited to, optical data storage, optical buffers,delays of optical data or pulses). Thus, in certain embodiments, theoptical device 10 is an optical data storage device, an optical buffer,or an optical delay device.

To determine the effect of the length of the FBG on the sensitivity,FIGS. 13A (1.064 μm) and 13B (1.55 μm) were generated showing the powersensitivity versus grating length for a fixed Δn of 1.5×10⁻⁴ inaccordance with certain embodiments described herein, calculatedassuming a lossless grating. For the slow-light transmission scheme(evaluated here again at λ₁ or equivalently λ′₁), the phase sensitivityfor operation at 1.064 μm scales approximately as L^(2.35). Thisdependence is not exactly universal, but close. For example, identicalsimulations (e.g., using the calculation scheme of Yariv and Yeh)carried out with a Δn=7.5×10⁻⁴ yielded a sensitivity that varied asL^(2.98). When Δn is further increased to 1.5×10⁻³, the sensitivitygrows as L^(2.97). These figures also depend on the exact spatialprofile of the index modulation (sinusoidal, square, etc.). Theconclusion is nevertheless that the sensitivity depends rapidly onlength. For operation of the slow-light transmission scheme at 1.55 μm,the phase sensitivity scales approximately as L^(2.89). For theslow-light reflection scheme (evaluated here again at λ_(a) orequivalently λ′_(a)) at 1.064 μm, the sensitivity also grows asL^(2.91), which is similar to the relationship seen in the slow-lighttransmission scheme. For the slow-light reflection scheme at 1.55 μm,the sensitivity also grows as L^(2.85).

In the above example of an FBG with λ_(Bragg)=1.064 μm, Δn of 1.5×10⁻⁴,and a length of 2 cm, the power sensitivity in the slow-lighttransmission mode was ˜8° C.⁻¹. FIG. 13A shows that when increasing thelength of this grating from 2 cm to 10 cm (an example high value deemedreasonable to use in a reflection grating, see FIG. 4), this powersensitivity increases to 877° C.⁻¹. These figures illustrate thedramatic improvement in sensitivity that can be obtained by increasingthe length and/or the index contrast of an FBG operated in theslow-light transmission mode in accordance with certain embodimentsdescribed herein. For the other example at 1.550 μm shown in FIG. 13B,for an FBG with a Δn of 1.5×10⁻⁴ and a length of 2 cm, the powersensitivity in the slow-light transmission mode was ˜2.7° C.⁻¹. FIG. 13Bshows that when increasing the length of this grating from 2 cm to 10cm, this power sensitivity increases to 243° C.⁻¹.

In certain embodiments the length L and index contrast Δn can beselected to provide a group index n_(g) greater than 10, greater than20, greater than 30, greater than 40, greater than 50, greater than 100,greater than 500, greater than 1,000, greater than 5,000, or greaterthan 10,000.

In one embodiment, the FBG is placed in one arm of a MZ interferometer,for example made of optical fiber, as depicted in FIG. 7. The MZinterferometer of certain embodiments is substantially balanced, exceptfor the purpose of biasing the two arms (e.g., π/2) and maximizing thesensitivity to a small phase change. When a small temperature change isapplied to the FBG, the phase of the signal traveling through the FBGchanges, whereas the phase of the signal traveling through the referencearm does not. When these two signals are recombined at the secondcoupler of the MZ interferometer, the signals interfere in a manner thatdepends on their relative phase shift, which is π/2+δφ, whereδφ=(dφ/dT)ΔT and dφ/dT is the sensitivity previously discussed andcalculated (for example, in FIGS. 10A and 10B). As a result of thisrelative phase shift, the signal output power at either port of the MZinterferometer changes by an amount proportional to δφ.

A fiber MZ interferometer typically has a minimum detectable phase (MDP)of the order of 0.1 to 1 μrad. As an example, for a MZ interferometerwith an MDP of 1 μrad, an index contrast of 0.015, and a grating lengthof 10 cm operating at 1.55 microns, the phase sensitivity is 4.8×10⁶rad/° C. Since the MPD is 1 μrad, this MZ-slow-light-sensor arrangementcan detect a temperature change as small as 2.1×10⁻¹³° C. This is, onceagain, nearly 5 million times greater than that of an optimizedreflection FBG of same length.

A further example of this principle is shown in FIGS. 14A and 14B, wherethe calculated minimum detectable temperature is plotted as a functionof FBG index contrast for an FBG used in the Bragg-reflection mode forλ_(Bragg)=1.064 μm and for λ_(Bragg)=1.55 μm, respectively, calculatedassuming a lossless grating. In this simulation, the grating length is 1cm for λ_(Bragg)=1.064 μm, and 2 cm for λ_(Bragg)=1.55 μm, and the MZinterferometer was taken to have an MDP of 1 μrad. As predicted inearlier simulations, in particular from FIGS. 11A and 11B, as the indexcontrast increases, the sensitivity drops, and therefore the minimumdetectable temperature increases. The same dependency is shown in FIGS.15A and 15B for an FBG of same length (and MDP) operated in theslow-light transmission mode in accordance with certain embodimentsdescribed herein, calculated assuming a lossless grating. In sharpcontrast to the Bragg-reflection mode, the minimum detectabletemperature decreases monotonically as the index contrast increases,eventually reaching exceedingly small values.

This example clearly illustrates the benefits provided by certainembodiments described herein over the prior Bragg-reflection mode ofoperation. First, for both slow-light configurations in accordance withcertain embodiments described herein, the sensitivity is considerablylarger. Second, for the slow-light transmission configuration inaccordance with certain embodiments described herein, the MZinterferometer does not need to be imbalanced, so both of its arms canhave extremely short lengths, and can therefore be fairly stable againsttemperature changes. Third, for both slow-light configurations inaccordance with certain embodiments described herein, the sensor canutilize a commercial laser as the source, unlike the reflection modeconfiguration of the prior art, which requires a broadband source in onecase (see, e.g., Kersey et al.) and its own laser in the second case(see, e.g., Koo and Kersey). The commercial laser can be chosen incertain embodiments to have an extremely narrow linewidth and low noiselimited by shot noise. In contrast, in the first case of theBragg-reflection configuration (e.g., FIG. 1), a broadband source ismuch noisier, which will add phase and intensity noise to the outputsignals at the detection, and further increase the MDP (and thus theminimum detectable temperature). In the second case of theBragg-reflection configuration (e.g., FIG. 2), the source is essentiallya custom laser including an FBG, which would require precise wavelengthstabilization in order to reduce laser line broadening and to keep thenoise low. This can be done, but again it requires a fair amount ofengineering, and it is more costly than commercial narrow-linewidthlasers, which are manufactured and sold in large quantities and benefitfrom an economy of scale.

This ability to detect a phenomenally small temperature is excessive formost applications. In practical applications, however, this highsensitivity can be traded for a shorter length. In the slow-lighttransmission mode example cited above for λ_(Bragg)=1.064 μm, the sensorhas a sensitivity of 2.2×10⁷ rad/° C. for a length of 10 cm. By reducingthis FBG length to 800 μm, or a factor of ˜125, according to theL^(2.88) dependence, the sensitivity will drop by a factor of ˜1.77×10⁶,down to 12.4 rad/° C. For the second slow-light transmission modeexample operating at 1.55 μm, the sensor has a phase sensitivity of4.8×10⁶ rad/° C. for a length of 10 cm. By reducing this FBG length to800 μm, the sensitivity will drop by a factor of approximately 1×10⁶,down to 4.8 rad/° C. These sensors still have about the same sensitivityas an optimized FBG used in Bragg-reflection mode (see FIG. 4), but itis only 800 μm long instead of 10 cm, and therefore it is considerablymore compact.

The above-described analysis was carried out for the case wheretemperature is the measurand. The same conclusions apply when themeasurand is another quantity, such as a longitudinal strain applieddirectly to the FBG.

By using slow light, both the strain sensitivity and the temperaturesensitivity are increased. Thus, one impact of a slow-light sensor inaccordance with certain embodiments described herein is that while it isa more sensitive strain sensor, it is also more sensitive to temperaturevariations. While the sensor can be stabilized against temperaturevariations in certain embodiments, such stabilization may not bedesirable. However, sensitivity and length can always be traded for oneanother. Hence, since the strain sensitivity and the temperaturesensitivity are enhanced in approximately the same proportions in theslow-light sensor of certain embodiments described herein, then thephysical length L of the grating can be reduced to bring the strainsensitivity and temperature sensitivity to the same levels as in abest-case Bragg-reflection FBG. The difference—and the benefit—of theslow-light configurations is that for equal sensitivity, the slow-lightFBG is considerably shorter, which can be important for manyapplications where compactness is critical. Any compromise of length andsensitivity is also possible, by which the slow-light sensor is designedso has to be somewhat shorter than a conventional reflection grating, aswell as more sensitive. In addition, the numerous engineering solutionsthat have been applied to discriminate between the change in strain andthe change in temperature applied to a grating are applicable in thepresent configurations of slow-light sensors. In particular, forexample, two gratings can be placed in parallel in the region wherestrain and temperatures are changing. One of the gratings is subjectedto the strain, but not the other, while both are subjected to the (same)temperature change. Comparison between the readings of the two sensorscan provide both the common temperature change and the strain changeapplied to one of the gratings.

Simulations also show that the linewidth of the source used tointerrogate an FBG operated in either of the slow-light modes in certainembodiments described herein is quite reasonable. FIG. 16 illustratesthis point with a plot of the power sensitivity dependence on thelinewidth of the laser for a grating with a length L=2 cm andΔn=1.5×10⁻⁴ operated at λ_(Bragg)=1.064 microns, calculated assuming alossless grating. As the laser linewidth increases, the sensitivity isconstant up to a linewidth of about 10⁻¹³ m. Above this value, thesensitivity starts decreasing, because the laser signal is probing awider spectral region that spans more than just the peak in the groupindex spectrum. In other words, some photons see a high group index (theones at frequencies at and around the peak of the group index spectrum),and others see a lower group index (the ones at frequencies detuned fromthis peak). This curve indicates that in order to obtain a maximumsensitivity, the laser linewidth is advantageously selected to be nolarger than about 10⁻¹³ m, or a frequency linewidth of 26 MHz. This is alinewidth that is readily available from a number of commercialsemiconductor lasers, for example Er—Yb-doped fiber lasers from NPPhotonics in Tucson, Ariz. A similar simulation carried out for an FBGof same length but large index contrast (Δn=1.5×10⁻³) yields a maximumvalue for the laser linewidth of about 2×10⁻¹⁵ m (530 kHz). Such laserlinewidth is also readily available commercially. The laser linewidththerefore can be decreased as the index contrast of the grating isincreased, or its length increased.

All simulations were carried out for FBGs with a Bragg wavelength ofeither 1064 nm (the primary wavelength of Nd:YAG lasers) or 1.55 μm.These wavelengths were selected because they are commonly used. However,the wavelength has no bearing on the general trends outlined in certainembodiments described herein. The properties of similar FBGs centered ata different wavelength, for example around 1.3 μm, do not differsubstantially from the properties presented herein, and they can bemodeled using the same equations presented and cited herein. Therelative benefits of the slow-light schemes in accordance with certainembodiments described herein over the Bragg-reflection described hereinremain substantially unchanged.

Optimization Process

The characteristics of the transmission and group index spectra of auniform grating can be uniquely determined by three parameters: indexmodulation, length, and loss. In a lossless grating, the case discussedabove, the group index can be enhanced by increasing the index contrastand the length indefinitely. In practice, when light travels though agrating, it encounters loss from scattering, which induces coupling intoa radiation mode. In the presence of loss, as the length of the gratingis increased, the light travels over a longer distance in the gratingand encounters correspondingly higher losses. This effect is enhancedwhen the group index of the FBG is large, because the light encountersmore loss as it travels many more times back and forth through thegrating. So for a given loss, as the grating length is increased, thegroup index first increases as described above. As the group indexfurther increases, the loss starts to limit the maximum number of roundtrips, much like it does in a Fabry-Perot interferometer, and the groupindex starts to decrease with any further increase in length. For agiven loss coefficient, there is consequently a grating length thatmaximizes the group index at the resonances. Similarly, as the lengthincreases, the loss also limits the transmission of the grating at theseresonances. When designing an FBG for slow light applications, it can bedesirable to carry out an optimization study, using for example theaforementioned model, to determine the optimal length of a grating givenits type of profile, index modulation, and loss. The loss coefficient ofthe FBG can be measured, using any number of standard techniques knownto persons with ordinary skill in the art. The measured power losscoefficient of FBGs ranges from 1 m⁻¹ in a Ge-doped grating (Y. Liu, L.Wei, and J. Lit, “Transmission loss of phase-shifted fiber Bragggratings in lossy materials: a theoretical and experimentalinvestigation,” Applied Optics, 2007) to more than 2 m⁻¹ in ahydrogen-loaded grating (D. Johlen, F. Knappe, H. Renner, and E.Brinkmeyer, “UV-induced absorption, scattering and transition losses inUV side-written fibers,” in Optical Fiber Communication Conference andthe International Conference on Integrated Optics and Optical FiberCommunication, 1999 OSA Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1999), paper ThD1, pp. 50-52).

This behavior is illustrated in FIG. 17 for an example uniform gratingoperated at 1.55 μm. This figure shows the dependence of the group indexand transmission at the first resonance (λ₁) on length in a strongGe-doped grating, e.g., a grating with a large index contrast(Δn=1.0×10⁻³ in this example) for a loss coefficient α varying between 1m⁻¹ and 2 m⁻¹. For a given loss coefficient, when the length of thegrating is short, the grating does not have sufficient periods and thegroup index is low. When the length is long, light encounters more lossas it travels through the grating and the group index decreases.Somewhere between these two limits, the group index is maximum. As shownin the example of FIG. 17, when the loss of the grating is 1 m⁻¹, thehighest group index is 84 at a grating length of 2.25 cm, and thetransmission is about 10%. When the loss increases to 2 m⁻¹, the highestgroup index decreases to 53 at a shorter optimum length of 1.8 cm; andthe transmission at this length is about the same (11%). Also asexplained above, the transmission decreases steadily as the length isincreased. In applications where the highest group index is desirableand the transmission is of less concern, operation at or in the vicinityof the optimum length is preferable. In applications where thetransmission is of more concern, a compromise can be made in order toachieve the highest group index possible without unduly reducing thetransmission. A slow-light FBG designer can also select other resonancewavelengths besides the first resonance wavelength modeled in thisexample.

FIG. 18 shows the same dependencies calculated for an even strongerexample FBG, fabricated in a hydrogen-loaded fiber. The value of Δn usedin this simulation is 0.01, a value reported for a grating written in ahydrogen-loaded fiber, and a loss coefficient of 2 m⁻¹. See P. J.Lemaire, R. M. Atkins, V. Mizrahi, and W. A. Reed, “High-pressureH₂-loading as a technique for achieving ultrahigh UV photosensitivityand thermal sensitivity in GeO₂-doped optical fibers,” Electron. Left.,1993. The FBG was assumed to be uniform for this calculation. Thehighest group index for this example occurs for a length of 0.37, and itis equal to 243. The transmission of the grating at that group index is12%.

Apodization also impacts the relationship between group index,transmission and length. Examples of two types of apodization, referredto herein as raised-Gaussian-apodized with zero-dc index change as typeA and Gaussian-apodized as type B are shown in FIG. 19A. See T. Erdogan,“Fiber grating spectra,” J. of Lightwave Technology, Vol. 15, pp.1277-1294, 1997. As illustrated in FIG. 19A, in type A, the indexprofile is modulated above and below some mean index value. In type B,the index profile is modulated strictly above some mean value. In bothtypes, the envelope of the index modulation can have any profile, e.g.,cosine or Gaussian. In the following simulations, for both type A andtype B FBGs the envelope is assumed to be a Gaussian with a full widthat half maximum (FWHM) labeled W. FIG. 19B illustrates the group indexspectrum calculated for a type-A apodized FBG with Δn=1.0×10⁻³, L=2 cm,a loss coefficient of 1.3 m¹, and W=2 L. The highest group index peakoccurs at the second resonance, but the value is smaller than that of auniform grating with the same index modulation, length and loss. Thereare four relevant parameters that control the slow-light behavior ofsuch an apodized grating: the maximum index modulation, the length, theloss coefficient, and the full width at half maximum (FWHM) W of theindex-profile envelope. Type B gratings produces asymmetric transmissionand group index spectra as shown in FIG. 19C and FIG. 19D forΔn=1.0×10⁻³, L=2 cm, a loss=1.3 m⁻¹, and W=2 L. The highest group indexoccurs at the first resonance peak at the shorter wavelength side. Withappropriate apodization, its value can be greater than that of a uniformgrating with the same index modulation and length. This can be achievedin certain embodiments by the optimization process described below.

For a given maximum index contrast Δn and a given loss coefficient, thetwo parameters that can be optimized to maximize the group index are thelength and the FWHM W. In a most general approach well known in anoptimization processes, one can carry out a straightforwardtwo-dimensional parametric study. As an example, FIG. 20 shows that therelationship between group index, transmission, and length for a type-Bapodized FBG in which the FWHM is equal to twice the length, the Δn is1.0×10⁻³, and the loss coefficient is 1 m⁻¹. The optimum length formaximum group index is 1.43 cm. At this length, the group index is ashigh as 178. The transmission is then 1.4%. Increasing the loss to 1.5m⁻¹ decreases the group index, and it reduces the optimum length, asexpected. However, the transmission increases slightly, to about 1.7%.Comparison to FIG. 17, which modeled a uniform FBG with the same indexcontrast, shows that for the same length and index modulation, anapodized grating of type B with the same length and index modulation asa uniform grating produces a higher group index and a lower transmissionthan a uniform grating. Tailoring the index profile of the FBG thereforecan have a significant impact on the design of certain embodimentsdescribed herein of an FBG used as a slow-light device.

The width of the apodization envelope also can play an important role inthe group index and transmission. When the FWHM is small, the effectivelength of the grating becomes small, and it leads to a lower groupindex. When the FWHM is large, the grating profile becomes similar to auniform grating. Therefore, in this limit, the group index andtransmission dependences on length converge to their respectivedependencies in the corresponding uniform grating. In FIG. 21, the groupindex and transmission are plotted against the FWHM W of the envelope inthe example case of a loss coefficient of 1 m⁻¹, a maximum indexcontrast of Δn=1.0×10⁻³, and a length of 1.43 cm. The optimum FWHM forthe maximum group index is 1.4 cm. At this FWHM, the group index is 200,which is even higher than in FIG. 20, but the power transmission is verylow. Again, a compromise can be made to decide which optimum length tochoose, depending on the application requirements, but curves such asFIGS. 20 and 21 clearly provide information which allows this choice tobe made.

The same optimization process can be applied to an apodizedhydrogen-loaded FBG of type B, as illustrated in FIG. 22. When the FWHMof the Gaussian envelope is equal to twice the length, the optimallength is 0.17 cm for a Δn of 1.0×10⁻² and a loss coefficient of 2 m⁻¹.The group index for this example reaches 744 and the power transmissionat this group index is 5%.

For the hydrogen-loaded FBG modeled in FIG. 22, assuming its length isselected to be 0.17 cm, FIG. 23 shows that the optimum FWHM is 0.17 cm.At this FWHM, the group index reaches 868, but the power transmission atthis point is very low. The FWHM that produces the highest group indexis approximately equal to the length of the grating in this case. Theadvantage of this particular FWHM is a very high group index, and thedrawback is a low power transmission. Here, as in all the other examplescited earlier, the tradeoff between the power transmission and groupindex is specific to each application.

Aside from uniform and apodized gratings, π-shifted grating is anothertype of a common grating profile that can produce slow-light. Aπ-shifted grating has a π phase shift located at the center of thegrating profile. This type of grating opens a narrow transmissionresonance at the Bragg wavelength, and it also broadens the transmissionspectrum. The lowest group velocity for this type of grating is nolonger located at the bandgap edge, but rather at the center of thebandgap λ_(π). This is illustrated in FIGS. 29A and 29B which show thetransmission and group index spectra calculated for a uniform π-shiftedgrating with Δn=2.0×10⁻⁴, L=2 cm, and zero loss.

These predictions were verified experimentally by measuring the groupdelay of light traveling through various FBGs with a Bragg wavelengthnear 1550 nm. Light that travels at a wavelength where a large groupindex occurs experiences a large group delay, proportional to the groupindex. The group delay was determined by measuring the time differencebetween the time of arrival of two signals of different wavelengths,both provided by the same tunable laser. The first wavelength waslocated far away (˜2 nm) from the bandgap edge of the FBG, such that thelight travels through the FBG at a normal group velocity. At this firstwavelength, the group index is very close to the phase index, which isitself very close to the refractive index of the material n₀, e.g.,about 1.45. The second wavelength was tuned to be close (within 200 pm)to the bandgap edge, where the group index, and therefore the groupdelay, are larger. The signals at the first wavelength and at the secondwavelength were both modulated in amplitude, at the same frequency,before entering the FBG. The difference between the group delay measuredat the two wavelengths provided a measure of the increase in group indexinduced by the FBG.

The experiment setup used for this measurement is depicted in FIG. 24.The beam from a tunable laser (Hewlett-Packard HP 81689A) was sentthrough an optical isolator and a polarization controller, then throughan amplitude modulator. The polarization controller was used to adjustthe state of polarization (SOP) of light entering the modulator andhence maximize the power transmitted by the modulator (whose operationhappened to be polarization dependent). A sinusoidal signal with afrequency f_(m) between 25 MHz to 100 MHz from a function generator wasfed into the modulator, which modulated the power of the laser signal.The sinusoidally modulated laser light was sent through the FBG undertest. The signal exiting the FBG was split into two using a 50/50 fibercoupler. One of the output signals was sent to a power meter to measureits power, and thus (when varying the laser wavelength) the transmissionspectrum of the FBG. The other beam was sent to a photodetector followedby a lock-in amplifier, which measured its phase. The first measurementwas conducted at a wavelength of 1548.000 nm, which is far enough awayfrom the bandgap edge (2 nm) that it does not experience slow light, andwas thus used as a reference signal. The laser was then tuned to aslow-light wavelength close to the bandgap edge, and the phasemeasurement was repeated. The difference in group delay between thefirst wavelength and the second wavelength was calculated from the phasechange Δφ measured between these two wavelengths using:

$\begin{matrix}{{\Delta \; \tau_{g}} = \frac{\Delta \; \varphi}{2\pi \; f_{m}}} & (8)\end{matrix}$

The group index at the second wavelength can be calculated from thedifferential group delay using:

$\begin{matrix}{n_{g} = {{\Delta \; \tau_{g}\frac{c}{L}} + n_{0}}} & (9)\end{matrix}$

Table 1 lists the commercial fiber Bragg gratings that have been tested.They were all manufactured by OE-Land in Canada. The table lists theirlengths, whether they were athermal gratings, and whether the indexprofile of the grating was uniform, according to the manufacturer. Italso lists the index contrast Δn of each grating (the peak value in thecase of a non-uniform FBG).

TABLE 1 Manu- Highest Grating facturer Uniformity Length Δn measuredn_(g) #1 OE Land Yes 2 cm 1.1 × 10⁻⁴ 3.7 #2 OE Land Yes 3 cm 1.1 × 10⁻⁴4.9 #3 OE Land No 2 cm 1.0 × 10⁻³ 69 #4 OE Land No 10 cm  1.0 × 10⁻³ 16#5 OE Land No 2 cm ~1.0 × 10⁻³   34

FIG. 25A shows, as an example, the measured transmission spectrum ofgrating #1, which has a length of 3 cm and a nominally uniform indexcontrast with a value specified by the vendor of about 1×10⁻⁴. Thetransmission spectrum exhibits the shape expected for a uniform grating,namely a narrow reflection peak centered at a Bragg wavelength (in thiscase λ_(Bragg)≈1549.948 nm), surrounded on both sides by oscillations ofdiminishing amplitude away from this peak. The solid curve in FIG. 25Ais the transmission spectrum calculated from theory for a uniform FBG.The index contrast is the only parameter that was adjusted to match thetheoretical curve to the experiment. This fit was used because the valueof the index contrast specified by the manufacturer was not accurateenough. The fitted value used to generate FIG. 25A, Δn=1.10×10⁻⁴, isclose to the vendor value. These simulations assumed zero loss. Thesecurves show that the presence, on both sides of the Bragg reflectionpeak, of transmission peaks with a transmission very close to 100%.

FIG. 25B shows the measured group index spectrum for the same grating(#1), as well as the theoretical spectrum calculated for the same Δn andlength as the solid curve in FIG. 25A. The light is slowest atwavelengths λ₁≈1549.881 nm and λ′₁≈1550.012 nm, which are symmetricallylocated with respect to λ_(Bragg), and also coincide with the first hightransmission peaks on either side of the FBG's bandgap. The highestmeasured value of the group index at these two wavelengths is ˜3.7,which is in excellent agreement with the predicted values. As in FIG.25A, there is a very good match between theory and experiment.

FBGs with a higher index contrast were tested, and as expected theyprovided a higher maximum group index. As an example, FIGS. 26A-26C showthe corresponding curves for grating #3, which had a length of 2 cm anda Δn of ˜1.0×10⁻³. FIG. 26A shows that the full measured transmissionspectrum is not symmetric about the Bragg wavelength, which isindicative of an apodized grating, as discussed above. FIG. 26B showsthe short-wavelength portion of the same measured transmission spectrum,magnified for convenience. Superposed to this measured spectrum is thefitted transmission spectrum predicted by the model, fitting fourparameters, namely the index contrast (Δn=1.042×10⁻³), length (L=20 mm),FWHM of Gaussian apodization (W=42 mm) and the loss coefficient (γ=1.6m⁻¹) to obtain the nominally best fit to the measured spectrum. Againthe agreement between measurements and theory is excellent. The fittedgroup index Δn is close to the value specified by the manufacturer, andthe loss coefficient is within the range of reported values for FBGs.FIG. 26C exhibits the measured group index spectrum for this grating, aswell as the predicted spectrum, calculated for the same fitted parametervalues as used in FIG. 26B. The maximum measured group index occurs atthe second slow-light peak (wavelength λ₂) and is 69, which is thehighest value reported to data in a fiber Bragg grating (the previousrecord was ˜5, as reported in J. T. Mok, C. Martijn de Sterke, I. C. M.Littler and B. J. Eggleton, “Dispersionless slow-light using gapsolitons,” Nature Physics 2, 775-780 (2006), and the highest reportedvalue in an optical fiber (the previous record was ˜10, as reported inC. J. Misas, P. Petropoulos, and D. J. Richardson, “Slowing of Pulses toc/10 With Subwatt Power Levels and Low Latency Using BrillouinAmplification in a Bismuth-Oxide Optical Fiber”, J. of LightwaveTechnology, Vol. 25, No. 1, January 2007). It corresponds to a groupvelocity of ˜4,350 km/s, by far the lowest value reported to date in anoptical fiber. The fit between the experimental and the theoreticalspectra are excellent. This value of n_(g)=69 was observed at the secondslow light peak from the Bragg wavelength (wavelength λ₂). The firstslow-light peak could not be measured because, as shown in FIG. 26B, thefirst peak (around a wavelength of λ₁=1549.692 nm) was too weak to bemeasured. The FBG transmission at the second peak was ˜0.5%. The groupindex measured at the third peak (wavelength λ₃) was only slightly lower(˜68), but the FBG transmission was significantly higher, about 8%. Thecorresponding values for the fourth peak (wavelength λ₄) were n_(g)≈43and a transmission of 32%. FIG. 26C also shows that the bandwidth of theslow-light peaks increases as the order i of the slow light peakincreases. This illustrates again that a given FBG can give a wide rangeof group index/group index bandwidth/transmission combination, which theuser can select based on the desired performance for the intendedapplication(s).

The last column in Table 1 summarizes the maximum n_(g) values measuredin the five gratings that were tested. In all cases except grating #4,the agreement between predicted and measured values was excellent. Inthe case of grating #4, the length was so long that the calculationfailed to converge and provide a reliable value.

The linewidths of the slow light peaks tend to decrease as the groupindex increases, e.g., as the index contrast or the length of thegrating are increased. To obtain the maximum benefit from a slow-lightFBG sensor, or from a slow-light FBG used for other purposes, a lasercan be selected with a linewidth that is smaller than the linewidth ofthe slow-light peak that is being used. If the linewidth of the laser isgreater than the linewidth of the slow-light peak, the laser photons atthe peak maximum experience maximum sensitivity, but photons detunedfrom the peak experience a lower sensitivity. The average sensitivitywill therefore be reduced. This can be illustrated with the laserlinewidths used in the measurements. For grating #1, which has a modestmaximum group index, the group-index linewidth of this slow-light peak(λ₁) was relatively broad, and its transmission and group index spectra(FIGS. 25A and 25B) could be probed with a laser of linewidth equal to˜1 pm. For grating #3, which has a much higher maximum group index, thegroup-index linewidths of the slow-light peaks (λ₂, λ₃, and λ₄) weremuch narrower, and its transmission and group index spectra (FIGS. 26Band 26C) were probed with a laser of linewidth equal to ˜0.8 fm (100 kHzin frequency). The linewidth of a slow-light peak can readily becalculated using the theoretical model described above, as has beenillustrated with FIGS. 25 and 26. From this linewidth prediction, it isstraightforward to compute the sensitivity of a sensor as a function oflaser linewidth. The sensor can alternatively be operated with a broaderlinewidth, the disadvantage being a lower sensitivity (as per FIG. 16)but the advantage being a greater stability against temperature changes.

Temperature affects the slow light spectrum. As the temperature of theFBG changes, its period Λ, effective mode index, and length all vary dueto a combination of thermal expansion and/or the temperature dependenceof the index of refraction dependence of the fiber materials. Theseeffects are well-known, and can readily be predicted usingwell-established mathematical models. As an example, the application ofthese basic effects to an FBG with L=2 cm, Δn1.5×10⁻⁴, andλ_(Bragg)=1.55 μm predicts a relative temperature sensitivity of thefirst transmission peak wavelengths (λ₁ and λ′₁) of approximatelyΔλ₁/λ₁=10 pm per ° C. If the FBG is used as a strain sensor for example,as the temperature of the grating changes, the sensitivity to strainwill generally vary because the transmission peak wavelengths vary withtemperature. This can be avoided in practice by controlling thetemperature of the FBG, to a degree that depends on the group index atthe wavelength of operation (the higher the group index, in general thetighter the temperature control). Alternatively, one can use an athermalFBG, commercial devices in which the inherent temperature dependence ofthe FBG spectrum has been partially compensated by properly packagingthe grating. Such devices are commercially available, for example fromOE Land or Teraxion in Canada.

Fiber Bragg gratings can be subject to phase or amplitude disorder,namely, random variations along the grating longitudinal axis z ineither the period of the grating or in the index contrast of thegrating. It is well known that the presence of such disorder alters theproperties of the FBG. In particular, generally such disorder results inbroadening of the reflection peak and reduction of its power reflectioncoefficient. Similarly, phase or amplitude disorder will result inmodification of the slow-light spectrum of an FBG, in particular ingeneral towards reducing the transmission and group index of theslow-light peaks. If these effects are deemed deleterious for theapplication considered, measures may be taken to minimize phase oramplitude disorder during the fabrication of a slow-light FBG.

FIG. 27 is a flowchart of an example method 1000 for using a fiber Bragggrating in accordance with certain embodiments described herein. Themethod 1000 comprises providing an FBG 20 comprising a substantiallyperiodic refractive index perturbation along a length of the FBG 20, asshown in operational block 1010. The FBG 20 has a power transmissionspectrum comprising a plurality of local transmission minima. Each pairof neighboring local transmission minima has a local transmissionmaximum therebetween. The local transmission maximum has a maximum powerat a transmission peak wavelength. The method 1000 also comprisesgenerating light having a wavelength between two neighboring localtransmission minima from a narrowband optical source, as shown inoperational block 1020 of FIG. 28. In certain embodiments, the generatedlight has a linewidth that is narrower than the linewidth of thetransmission peak. The method 1000 further comprises transmitting afirst portion 33 a of light along a first optical path 31 extendingalong and through the length of the FBG 20 in operational block 1030,and transmitting a second portion 33 b of light along a second opticalpath 32 in operational block 1040. In certain embodiments in which theFBG is used in an optical sensor, the method 1000 further comprisesdetecting the first portion 33 a, the second portion 33 b, or both thefirst and second portions 33 a 33 b with an optical detector 30 inoperational block 1050.

In certain embodiments of the method 1000, the substantially periodicrefractive index perturbation has a constant period along the length ofthe FBG 20. In certain other embodiments, the substantially periodicrefractive index perturbation has a period which varies along the lengthof the FBG 20 such that the FBG 20 is a chirped grating. In someembodiments, the substantially periodic refractive index perturbationhas an amplitude which varies along the length of the FBG 20 such thatthe FBG 20 is an apodized grating.

In certain embodiments of the method 1000, the method 1000 furthercomprises recombining and transmitting the first and second portions 33a 33 b to an optical detector 40. For example, in certain embodiments,the method 1000 comprises providing a first fiber coupler 51 in opticalcommunication with the narrowband light source 30, the first opticalpath 31, and the second optical path 32; and providing a second fibercoupler 52 in optical communication with the first optical path 31 andthe second optical path 32. In these embodiments, the method 1000includes splitting the light generated by the narrowband optical source30 by the first fiber coupler 51 into the first portion 33 a and thesecond portion 33 b. Thus, in these embodiments, recombining andtransmitting are accomplished by the second fiber coupler 52. Also, inthese embodiments, detecting 1050 comprises detecting a phase differencebetween the first portion 33 a and the second portion 33 b. In certainembodiments, the first optical path 31 and the second optical path 32form a nominally balanced Mach-Zehnder interferometer.

In certain embodiments, the phase difference is indicative of an amountof strain applied to the FBG 20. In some embodiments, the phasedifference is indicative of a temperature of the FBG 20.

In certain embodiments of the method 1000, transmitting 1040 a secondportion 33 b of light along a second optical path 32 comprisesreflecting the second portion 33 b from the FBG 20. In theseembodiments, detecting 1050 can comprise detecting an optical power ofthe first portion 33 a, the second portion 33 b, or both the first andsecond portions 33 a 33 b. In some embodiments, the detected opticalpower is indicative of an amount of strain applied to the FBG 20. Insome embodiments, the detected optical power is indicative of atemperature of the FBG 20. In certain embodiments of the method 1000,the first portion 33 a transmitted along the FBG 20 has a first groupvelocity less than a second group velocity of light having a wavelengthoutside a reflected range of wavelengths transmitted along the FBG 20.In some of these embodiments, the ratio of the first group velocity tothe second group velocity is equal to or less than ⅓. In someembodiments, the ratio of the first group velocity to the second groupvelocity is equal to or less than 1/10.

FIG. 28 is a flowchart of another embodiment of a method 2000 for usinga fiber Bragg grating in accordance with certain embodiments describedherein. The method 2000 comprises providing an FBG 20 comprising asubstantially periodic refractive index perturbation along a length ofthe FBG 20, as shown in operational block 2010. In certain embodimentsof the method 2000, the method 2000 comprises generating light having awavelength from a narrowband optical source 30, as shown in operationalblock 2020. In certain embodiments, the wavelength is in the vicinity ofa slow-light peak of the FBG 20. The method 2000 further comprisestransmitting a first portion 33 a of light along a first optical path 31extending along and through the length of the FBG 20 with a groupvelocity such that a ratio of the speed of light in vacuum to the groupvelocity is greater than 5 in operational block 2030, and transmitting asecond portion 33 b of light along a second optical path 32 inoperational block 2040. In certain embodiments in which the FBG 20 isused for optical sensing, the method 2000 further comprises detectingthe first portion 33 a, the second portion 33 b, or both the first andsecond portions 33 a 33 b with an optical detector 40 in operationalblock 2050.

Optimization with Figures of Merit

The operation of certain embodiments of a novel type of fiber sensorutilizing a fiber Bragg grating (FBG) as the sensing element isdescribed herein and in U.S. patent application Ser. No. 12/792,631,filed on Jun. 2, 2010, which is incorporated in its entirety byreference herein. A difference with other FBG sensors reported to datein the literature is that certain embodiments described herein sensewith slow light. Slow light can be excited in the FBG by launching lightof a particular wavelength into the FBG. In certain embodimentsimplementing this concept, this wavelength can be selected in thevicinity of one of the high-transmission peaks that exist for certainFBGs on the edges of the FBG's bandgap. On the short-wavelength side ofthe bandgap as shown in FIGS. 9A and 9E, these wavelengths can belabeled λ_(j), where j is an integer greater or equal to 1, the peak j=1being the peak closest to the bandgap. On the long-wavelength side ofthe bandgap, these wavelengths can be labeled λ′_(j), where j is aninteger greater or equal to 1, the peak j=1 being the peak closest tothe bandgap. At these wavelengths, the FBG can support slow light,characterized by a group velocity that can be substantially lower thanthat of light normally traveling in an optical fiber. This low groupvelocity ν_(g) can be characterized by a high group index n_(g)=c/ν_(g),where c is the speed of light in vacuum.

When a perturbation (e.g., a strain) to be sensed is applied to a phasesensor in accordance with certain embodiments, the resultingperturbation of the phase of the light traveling through the sensor isproportional to the reciprocal of the group velocity. Consequently, inthese embodiments, operating an FBG in the vicinity of one of thesetransmission peaks can result in an increased sensitivity to ameasurand. This can be true in particular, but not limited to,temperature, strain, displacement, and relative rotation. In certainembodiments of FBG sensors utilizing slow light, the sensitivity to themeasurand can therefore scale like the group index. Thus, in certainembodiments, with everything else being the same, the higher the groupindex, or the slower the group velocity, the higher the sensitivity.

As disclosed in certain embodiments described herein and in U.S. patentapplication Ser. No. 12/792,631, light generally can have the lowestgroup velocity at the first (j=1) peak. For example, certain embodimentsof a uniform FBG can have the lowest group velocity on both sides of thebandgap, namely at both λ₁ and λ′₁. As another example, certainembodiments of an apodized FBG described herein and in U.S. patentapplication Ser. No. 12/792,631, can have the lowest group velocity onthe short-wavelength side of the bandgap, namely at λ₁. Owing to theasymmetry of the spectral response of these particular apodizedgratings, certain embodiments may exhibit little to substantially nopronounced high transmission peaks on the long-wavelength side of thebandgap.

FIG. 30 shows an example transmission spectrum calculated for an exampleapodized grating. The example transmission spectrum was calculated foran apodized grating with a length L=1.2 cm, a loss of 1 m⁻¹, and aGaussian index profile envelop with a peak index modulation Δn=1.04×10⁻³and a full width at half maximum (FWHM) W=0.98 cm. Its peak indexmodulation is 0.06 cm away from the center position of the grating at0.6 cm. With the apodization shift the n_(g) in peak No. 1 in FIG. 30 is218.6 and in peak No. 2 is 126.8. Without the shift, the n_(g) in peakNo. 1 is 225.2 and in peak No. 2 is 148.4. Shifting the peak indexmodulation from the center position of the grating will reduce then_(g), but in this case, the change in group index is not significantbecause the shift is small compared to the length of the grating. Forclarity, only the short-wavelength side of the spectrum is shown. Asdescribed above for certain embodiments, the transmission spectrumexhibits sharp peaks on the edge of the bandgap. In FIG. 30, λ_(B)points to the location of the Bragg wavelength of the FBG, locatedoutside of the figure at a wavelength of 1550.176 nm in this example. Inthis example, the transmission peaks have peak transmission values thatincrease with the peak number j. In this example, the transmission peaksalso broaden as the peak number increases.

FIG. 30 also shows the calculated group index spectrum of the exampleapodized grating. Again, for clarity, only the short-wavelength side isshown. As discussed for certain embodiments described herein and in U.S.patent application Ser. No. 12/792,631, this spectrum also can exhibitsharp resonances, centered at wavelengths that fall very close to thewavelengths λ_(j) and λ′_(j) of the transmission peaks. In certainembodiments, the first peak (j=1) exhibits the highest n_(g), e.g., theslowest group velocity. In these embodiments, the maximum value of thegroup index of subsequent slow-light peaks decreases as the number ofthe peak (j) increases.

Based on these concepts, two general classes of sensors in accordancewith certain embodiments are disclosed herein and in U.S. patentapplication Ser. No. 12/792,631. In certain embodiments of the firstclass, referred to as the transmission mode, the FBG is placed in one ofthe arms of a nominally balanced Mach-Zehnder (MZ) interferometer, andthe interferometer can be probed at a wavelength in the vicinity of aslow-light peak of the FBG. An example diagram of this approach isdescribed above and shown in FIG. 7. In certain embodiments of thesecond class of sensors, the wavelength can be selected to fall oneither side of a slow light peak, at or in the vicinity of thewavelengths where the slope of the group index spectrum is maximum.These sensors can be then used in the reflection mode, as describedabove and shown in FIG. 8.

Based on the proportionality of the sensitivity to group index alone,one may be inclined to conclude that to achieve the highest possiblesensitivity, one may probe the sensor at a wavelength in the vicinity ofthe maximum of the first slow-light peak. However, for certainembodiments, this may not necessarily the case, as will be describedbelow.

a. TRANSMISSION CONFIGURATION

FIG. 31 shows a diagram of an example implementation of an apparatusutilizing an FBG used in the slow-light transmission mode in accordancewith certain embodiments described herein. As shown in FIG. 31, theoptical device 110 can comprise an FBG 120 comprising a substantiallyperiodic refractive index modulation along a length of the FBG 120. TheFBG 120 can have a power transmission spectrum comprising a plurality oflocal transmission maxima, the local transmission maxima each having amaximum power at a transmission peak wavelength. The FBG 120 has a groupindex spectrum as a function of wavelength. The optical device 110 cancomprise a narrowband optical source 130 in optical communication with afirst optical path 131 and a second optical path 132. The narrowbandoptical source 130 can be configured to generate light, which can beconfigured to be split into a first portion 133 a and a second portion133 b. The first portion 133 a can be transmitted along the firstoptical path 131 extending along and through the length of the FBG 120at a group velocity. The light can have a wavelength at or in thevicinity of a wavelength at which the product of the group indexspectrum and the square root of the power transmission spectrum is at amaximum value (e.g., higher than for any other wavelengths in thevicinity of an edge of the bandgap of the FBG 120).

In certain embodiments, the FBG 120 can be similar to the FBG 20 asdescribed herein. For example, the FBG 120 can be fabricated by exposingthe core of an optical fiber to a spatially modulated UV beam. The indexmodulation can take any number of spatial distributions. The opticalfiber can be a conventional single-mode fiber or a multimode fiber. Theoptical fiber can be doped with special elements such that exposure tospatially varying light can induce a desired modulation in therefractive index. The spatially periodic refractive index modulation inthe FBG 120 can have a constant period along the length of the FBG 120,as in uniform gratings, can have a period that varies along the lengthof the FBG 120, as in chirped gratings, or can have the amplitude of theindex modulation vary along the length of the FBG 120, as in apodizedgratings. As shown in FIGS. 9A and 9E, the FBG 120 can have a powertransmission spectrum comprising a plurality of local transmissionmaxima, the local transmission maxima each having a maximum power at atransmission peak wavelength.

In certain embodiments, the narrowband optical source 130 can be similarto the narrowband optical source 30 described herein. For example, thenarrowband optical source 130 can comprise a semiconductor laser, or afiber laser, e.g., Er-doped fiber laser with a wavelength range betweenapproximately 1530 nm and 1565 nm. As another example, the narrowbandoptical source 130 can comprise a Nd:YAG laser with a wavelength of1064.2 nm. In some embodiments, the narrowband optical source 130 canhave a narrow linewidth, e.g., less than or equal to 10⁻¹³ m.

The narrowband optical source 130 can be in optical communication with afirst optical path 131 and a second optical path 132. The lightgenerated by the narrowband optical source 130 can be split into a firstportion 133 a and a second portion 133 b. In certain embodiments, thefirst portion 133 a can be transmitted along the first optical path 131extending along and through the length of the FBG 120 at a groupvelocity. In certain embodiments, the second portion 133 b can betransmitted along the second optical path 132 not extending along thelength of the FBG 120. In certain embodiments, the first optical path131 can be different from the second optical path 132, as shown in FIG.31. In other embodiments, the first optical path 131 and the secondoptical path 132 may at least partially overlap one another. The firstoptical path 131 and the second optical path 132 may transverse freespace or various optical elements. For example, the first optical path131 and/or the second optical path 132 may transverse an opticalelement, e.g., a fiber coupler as will be discussed below. In certainembodiments, the light generated by the narrowband optical source 130can have a wavelength at or in the vicinity of a wavelength at which theproduct of the group index and the square root of the power transmissionis the highest, as also will be discussed more fully below.

In certain embodiments, the optical device 110 can comprise at least oneoptical detector 140. The optical detector 140 can be configured to bein optical communication with the FBG 120. The optical detector 140 canbe configured to receive and detect an optical power of the firstportion 133 a of light, the second portion 133 b of light, or both thefirst portion 133 a and the second portion 133 b of light. In FIG. 31,the optical detector 140 can receive and detect both the first portion133 a and the second portion 133 b of light. In certain embodiments, theoptical detector 140 can be similar to the optical detector 40 describedherein. For example, the optical detector 140, can be a general purposelow-noise photodetector.

In certain embodiments, the optical device 110 can comprise a firstfiber coupler 151 in optical communication with the narrowband lightsource 130, the first optical path 131, and the second optical path 132.As show in FIG. 31, the light generated by the narrowband optical source130 can be split by the first fiber coupler 151, e.g., a 3-dB fibercoupler, into the first portion 133 a and the second portion 133 b. Thefirst portion 133 a can be transmitted along the first optical path 131.The second portion 133 b can be transmitted along the second opticalpath 132. The first portion 133 a can propagate along the FBG 120 whilethe second portion 133 b may not substantially interact with the FBG120. The first portion 133 a in this embodiment can include informationregarding the perturbation of the FBG 120, while the second portion 133b in this embodiment can remain unaffected by the perturbation.

The optical sensor 110 further can comprise a second fiber coupler 152,e.g., a 3-dB fiber coupler, in optical communication with the firstoptical path 131 and the second optical path 132. The first portion 133a and the second portion 133 b can be recombined by the second fibercoupler 152 and transmitted to at least one optical detector 140. Asdiscussed herein, this recombination can allow the first portion 133 aand the second portion 133 b to interfere with one another, producing acombined signal that can contain information regarding the phasedifference between the first portion 133 a and the second portion 133 b.As shown in FIG. 31, the optical detector 140 can comprise a singleoptical detector at one of the output ports of the second fiber coupler152. In certain other embodiments, as schematically illustrated by FIG.7, the optical detector 140 can comprise a first optical detector 40 aat one output port of the second fiber coupler 52 and a second opticaldetector 40 b at the other output port of the second fiber coupler 52.In certain embodiments, the phase difference can be indicative of anamount of strain applied to the FBG 120. In certain other embodiments,the phase difference can be indicative of the temperature of the FBG120.

As described above, the light generated by the narrowband optical source130 can have a wavelength at or in the vicinity of a wavelength at whichthe product of the group index and the square root of the powertransmission is the highest. For example, as shown in FIG. 31, theoptical device 110 can be an optical sensor in a transmission mode ofoperation, e.g., a MZ interferometer as described herein. The signal atthe output of the optical device, e.g., a MZ interferometer, can be thecoherent sum of the field E₁ transmitted into port 152 a by the firstoptical path 131, and the field E₂ transmitted into port 152 b by thesecond optical path 132. These fields can be written as:

$\begin{matrix}{{E_{1} = {E_{0}\sqrt{1 - \eta}{\exp \left( {\; \varphi_{1}} \right)}t_{1}\sqrt{1 - \eta}}}{E_{2} = {E_{0}\sqrt{\eta \;}{\exp \left( {\; \frac{\pi}{2}} \right)}{\exp \left( {\; \varphi_{2}} \right)}t_{2}\sqrt{\eta}{\exp \left( {\; \frac{\pi}{2}} \right)}}}} & (10)\end{matrix}$

where E₀ is the field produced by the narrowband optical source 130 andincident on the first fiber coupler 151, √η is the field couplingcoefficient of the first optical coupler 151, or equivalently η is thepower coupling coefficient of the first optical coupler 151, φ₁ and φ₂are the phase accumulated by light propagation through the first opticalpath 131 and the second optical path 132, respectively, and t₁ and t₂are the field transmission of the first optical path 131 and the secondoptical path 132, respectively. The exp(iπ/2) phase terms account forthe well-known π/2 phase shift that light picks up when it is coupledacross a coupler. At the upper output port 152 a of the optical device110, the field is given as the coherent sum of E₁ and E₂, and the outputpower P_(out) is proportional to the square of the modulus of this totalfield. Hence:

$\begin{matrix}\begin{matrix}{P_{{out}\;} = {P_{0}{{{\sqrt{1 - \eta}{\exp \left( {\; \varphi_{1}} \right)}t_{1}\sqrt{1 - \eta}} + {\sqrt{\eta}{\exp \left( {\; \varphi_{2}} \right)}t_{2}\sqrt{\eta}{\exp ({\pi})}}}}^{2}}} \\{= {P_{0}{{{\left( {1 - \eta} \right)t_{1}} - {\eta \; t_{2}{\exp \left( {\; \Delta \; \varphi} \right)}}}}^{2}}}\end{matrix} & (11)\end{matrix}$

where P₀ is the power incident on the first fiber coupler 151 of theoptical device 110, and Δφ=φ₂−φ₁ is the difference between the phasesexperienced by the two signals in the two optical paths 131, 132.Expanding the square in the last equality of Equation 11 gives:

P _(out) =P ₀((1−η)² t ₁ ²+η² t ₂ ²)−2P ₀η(1−η)t ₁ t ₂ cos(Δφ)  (12)

The first term in the right hand side of Equation 12 is a DC termindependent of the phase and of the phase perturbation applied to theFBG 120. The second term contains the interference term between the twowaves, and therefore the one that can contain important phaseinformation.

In accordance with certain embodiments, when a perturbation δψ isapplied to the FBG 120 in the optical device 110 of FIG. 31, thisperturbation can induce a change in the phase of the signal propagatingthrough the FBG 120. This phase perturbation can be proportional to theperturbation (for small enough perturbations) and as seen above, to thegroup index of the light in the FBG 120. It may therefore be written as:

$\begin{matrix}{{\delta \; \varphi} \propto {\frac{2\pi}{\lambda}n_{g}\delta \; \psi}} & (13)\end{matrix}$

The phase difference Δφ in Equation 12 is the sum of a constant term,which is the built-in phase difference between the two optical paths131, 132, and this phase perturbation δφ. When this built-in phasedifference is selected to be π/2 (modulo π), the output power P_(out)depends maximally on a small perturbation δψ. The phase-dependentportion of the output power (second term in the right hand side ofEquation 12) can then be written as:

$\begin{matrix}\begin{matrix}{{P_{out}({\delta\varphi})} = {{- 2}P_{0}{\eta \left( {1 - \eta} \right)}t_{1}t_{2}{\cos \left( {\frac{\pi}{2} + {\delta \; \varphi}} \right)}}} \\{= {2P_{0}{\eta \left( {1 - \eta} \right)}t_{1}t_{2}{\sin \left( {\delta \; \varphi} \right)}}}\end{matrix} & (14)\end{matrix}$

In certain embodiments, if the phase perturbation is small (as whenattempting to measure extremely small perturbations applied to the FBG120), sin(δφ)≈δφ, and Equation 14 becomes:

$\begin{matrix}{{P_{out}\left( {\delta \; \varphi} \right)} \approx {2P_{0}{\eta \left( {1 - \eta} \right)}t_{1}t_{2}\delta \; \varphi} \propto {2P_{0}{\eta \left( {1 - \eta} \right)}t_{1}t_{2}\; \frac{2\pi}{\lambda}n_{g}\delta \; \psi}} & (15)\end{matrix}$

using Equation 13 to replace δφ in the rightmost side of the equation.

Thus, in certain embodiments, the output power, which is the signalprovided by the optical device 110, e.g., MZ interferometer, as a resultof the perturbation applied to the FBG 120, is proportional toη(1−η)t₁t₂n_(g). To maximize this signal, and therefore the sensitivityof the optical device 110 in accordance with certain embodimentsdescribed herein, one can first maximize the product η(1−η). This can beachieved when η=0.5. The sensitivity of the optical device 110, e.g., aMZ interferometer, is maximum when the first fiber coupler 151 and thesecond fiber coupler 152 have a 50% power coupling coefficient. Thesecond item that can be maximized to maximize the output power is theproduct t₁t₂n_(g). This can be achieved by first maximizing thetransmission t₂ of the second optical path 132, e.g., the reference armof a MZ interferometer. The second step is to maximize the productt₁n_(g). The field transmission t₁ of the first optical path 131 can bemore conveniently expressed as √T₁, where T₁ is the power transmissionof the first optical path 131. The sensitivity of the optical device ofFIG. 31 in accordance with certain embodiments can therefore bemaximized not simply when n_(g) is highest, but when the productn_(g)√T₁ is the highest. The relevant figure of merit that is maximizedto maximize the sensitivity of this optical device 110 is thereforeF_(t)=n_(g)√T₁.

FIG. 32 shows an example figure of merit, which is the product of thesquare root of the transmission spectrum of FIG. 30 by the group indexspectrum of FIG. 30. As discussed earlier, both T₁ (plotted in FIG. 30)and n_(g) (plotted in FIG. 30) depend strongly on wavelength. The figureof merit F_(t) is therefore also strongly wavelength dependent, as shownin FIG. 32. FIG. 32 shows that in this particular example of an apodizedFBG, the figure of merit (or sensitivity) is not maximum at thewavelength of the first slow-light peak. As shown in FIG. 30, at thefirst slow-light peak, n_(g) is highest but T₁ is very weak for thisexample. On the other hand, at the second peak, n_(g) is only marginallysmaller, but T₁ is significantly higher, hence F_(t) is higher at thesecond peak than at the first peak for this example. In this example,for subsequent slow-light peaks, n_(g) decreases more than T₁ increases,so the figure of merit at these other peaks is lower than at the secondpeak. The net result is that operating at the second peak (wavelengthλ₂) yields in this particular example the highest figure of merit, andhence the most sensitive optical device 110. Plotting figures such asFIGS. 30 and 32 can be helpful to figure out, for a particular FBG 120,the optimum wavelength of operation for maximum sensitivity.

Plotting figures, such as FIGS. 30 and 32 may not be necessary incertain embodiments. For example, in certain embodiments using a uniformgrating, the peaks in the transmission spectrum all have a maximum valueof 1. The wavelengths of maximum sensitivity are dictated by the groupindex spectrum, and they coincide approximately with the wavelengthsλ_(j) and λ′_(j). The wavelengths that provide the highest sensitivityin these embodiments are therefore the ones that maximize the groupindex, which are λ₁ and λ′₁. However, in order to determine thebandwidth of certain embodiments of the slow-light sensor, orequivalently the maximum perturbation it can detect, it can beadvantageous to plot the spectrum of the figure of merit F, since itcontains the spectral, and therefore the bandwidth information of thedetected signal.

b. REFLECTION CONFIGURATION

Certain embodiments described herein utilize an FBG used in theslow-light reflection mode, e.g., as shown in FIG. 8. The optical device10 can comprise a narrowband optical source 30 in optical communicationwith a first optical path 31 and a second optical path 32. Thenarrowband optical source 30 can be configured to generate light, whichis configured to be split by the FBG 20 into a first portion 33 a and asecond portion 33 b. The light can have a wavelength at or in thevicinity of a wavelength at which the slope of the product of the groupindex spectrum and one minus the power transmission spectrum as afunction of wavelength is a maximum value (e.g., higher than for anyother wavelengths in the vicinity of an edge of the bandgap of the FBG20). In certain embodiments, the light can have a wavelength at or inthe vicinity of a wavelength at which the slope of the product of thegroup index spectrum and the power transmission spectrum as a functionof wavelength is a maximum value (e.g., higher than for any otherwavelengths in the vicinity of an edge of the bandgap of the FBG 20).

In addition, as discussed above, certain embodiments of the opticaldevice 10 shown in FIG. 8 further can comprise at least one opticaldetector 40 a and/or 40 b. The optical device 10 of FIG. 8 also caninclude at least one fiber coupler 51.

In certain embodiments, the light generated by the narrowband opticalsource 30 can be split by the FBG 20 into a first portion 33 a and asecond portion 33 b. In certain embodiments, at least one opticaldetector 40 a and/or 40 b can be configured to receive the first portion33 a, the second portion 33 b, or both the first and second portions 33a, 33 b of light.

In certain embodiments, the FBG 20 can be interrogated with a narrowbandoptical source 30, the light generated by the narrowband optical source30 can be split by the FBG 20 into the first portion 33 a and the secondportion 33 b. The wavelength of the light interrogating the FBG 20 canbe at or in the vicinity of a wavelength at which the slope of theproduct of the group index and one minus the power transmission as afunction of wavelength is a maximum, discussed more fully below. Incertain embodiments, the wavelength of the light interrogating the FBG20 can be at or in the vicinity of a wavelength at which the slope of aproduct of the group index and the power transmission as a function ofwavelength is a maximum, discussed more fully below.

As discussed above, when an external perturbation is applied to the FBG20, the reflection peak can shift in wavelength. This shift of λ_(Bragg)can result in a change in the first portion 33 a of light transmitted bythe FBG 20 and in the second portion 33 b of light reflected by the FBG20, for example, in the power of the reflected light at the wavelengthof the light incident on the FBG 20.

In the case of the reflection mode of operation (e.g., FIG. 8), theoutput power of the optical device 10 in accordance with certainembodiments is proportional to n_(g), as in the transmission mode, e.g.,FIG. 31, times the power reflection coefficient of the FBG 20, which isR₁=1−T₁ when the output power is the power reflected by the FBG 20. Therelevant figure of merit is then F_(r)=(1−T₁)n_(g). When the outputpower is the power transmitted by the FBG 20, the relevant figure ofmerit is F′_(r)=T₁n_(g). Plots similar to FIG. 32 can be plotted toobtain the spectrum of either F_(r) or F′_(r), depending on which outputis measured as the output of the optical device 10. The wavelength ofoperation that maximizes the sensitivity is then given by the wavelengthat which the slope of the figure of merit is maximum.

c. FURTHER EXAMPLE EMBODIMENTS

FIG. 33A is a flowchart of an example method 3000 for using a fiberBragg grating in accordance with certain embodiments described herein.The method 3000 comprises providing an FBG 20 comprising a substantiallyperiodic refractive index perturbation along a length of the FBG 20, asshown in operational block 3010. The FBG 20 can have a powertransmission spectrum comprising a plurality of local transmissionmaxima. The local transmission maxima each has a maximum power at atransmission peak wavelength. The method 3000 also comprises generatinglight, as shown in operational block 3020 of FIG. 33A. The light can begenerated by a narrowband optical source 30. In certain embodiments, thenarrowband optical source 30 is in optical communication with a firstoptical path 31 and a second optical path 32. The light is also splitinto a first portion 33 a of light and a second portion 33 b of light.The method 3000 further can comprise transmitting a first portion 33 aof light along a first optical path 31 extending along and through thelength of the FBG 20 at a group velocity in operational block 3030. Incertain embodiments, the light can have a wavelength at or in thevicinity of a wavelength at which the product of the group index and thesquare root of the power transmission is at a maximum value (e.g.,higher than for any other wavelengths in the vicinity of an edge of thebandgap of the FBG 20).

Certain embodiments of the method 3000 further can comprise receivingthe first portion 33 a, the second portion 33 b, or both the firstportion 33 a and the second portion 33 b of light with an opticaldetector 40; and detecting an optical power of the first portion 33 a,the second portion 33 b, or both the first portion 33 a and the secondportion 33 b of light. The method 3000 further can comprise transmittingthe second portion 33 b of light along the second optical path 32. Thesecond optical path 32 might not extend along and through the FBG 20. Invarious embodiments, the method 3000 further can comprise providing afirst fiber coupler 51 in optical communication with the narrowbandoptical source 30, the first optical path 31, and the second opticalpath 32. Furthermore, the method 3000 can comprise providing a secondfiber coupler 52 in optical communication with the optical detector 40,the first optical path 31, and the second optical path 32.

In certain embodiments of the method 3000, the method 3000 further cancomprise recombining and transmitting the first and second portions 33a, 33 b of light to the optical detector 40. In these embodiments, themethod 3000 can include splitting the light generated by the narrowbandoptical source 30 by the first fiber coupler 51 into the first portion33 a and the second portion 33 b. Thus, in these embodiments,recombining and transmitting can be accomplished by the second fibercoupler 52. Also, in these embodiments, detecting can comprise detectinga phase difference between the first portion 33 a and the second portion33 b. In certain embodiments, the first optical path 31 and the secondoptical path 32 can form a nominally balanced Mach-Zehnderinterferometer. In certain embodiments, the phase difference can beindicative of an amount of strain applied to the FBG 20. In otherembodiments, the phase difference can be indicative of a temperature ofthe FBG 20.

In certain embodiments of the method 3000, the substantially periodicrefractive index perturbation can have a constant period along thelength of the FBG 20. In certain other embodiments, the substantiallyperiodic refractive index perturbation can have a period which variesalong the length of the FBG 20 such that the FBG 20 is a chirpedgrating. In some embodiments, the substantially periodic refractiveindex perturbation can have an amplitude which varies along the lengthof the FBG 20 such that the FBG 20 is an apodized grating.

FIG. 33B is a flowchart of another embodiment of a method 4000 for usinga fiber Bragg grating in accordance with certain embodiments describedherein. The method 4000 can comprise providing an FBG 20 comprising asubstantially periodic refractive index perturbation along a length ofthe FBG 20, as shown in operational block 4010. In certain embodimentsof the method 4000, the method 4000 can comprise generating light havinga wavelength, as shown in operational block 4020. The light can begenerated from a narrowband optical source 30. In certain embodiments,the narrowband optical source 30 can be in optical communication with afirst optical path 31 and a second optical path 32. The light can alsosplit into a first portion 33 a of light and a second portion 33 b oflight. The method 4000 further can comprise transmitting a first portion33 a of light along a first optical path 31 extending along and throughthe length of the FBG 20 at a group velocity in operational block 4030.In certain embodiments, the light has a wavelength at or in the vicinityof a wavelength at which the slope of a product of the group index andone minus the power transmission as a function of wavelength is amaximum (e.g., a maximal value compared to the values of this quantityfor any other wavelengths in the vicinity of an edge of the bandgap ofthe FBG 20).

Certain embodiments of the method 4000 further can comprise receivingthe first portion 33 a, the second portion 33 b, or both the firstportion 33 a and the second portion 33 b of light with an opticaldetector 40; and detecting an optical power of the first portion 33 a,the second portion 33 b, or both the first portion 33 a and the secondportion 33 b of light. The method 4000 further can comprise reflectingthe second portion 33 b of light along the second optical path 32.

In certain embodiments of the method 4000, the substantially periodicrefractive index perturbation has a constant period along the length ofthe FBG 20. In certain other embodiments, the substantially periodicrefractive index perturbation has a period which varies along the lengthof the FBG 20 such that the FBG 20 is a chirped grating. In someembodiments, the substantially periodic refractive index perturbationhas an amplitude which varies along the length of the FBG 20 such thatthe FBG 20 is an apodized grating.

FIG. 33C is a flowchart of another embodiment of a method 5000 for usinga fiber Bragg grating in accordance with certain embodiments describedherein. The method 5000 can comprise providing an FBG 20 comprising asubstantially periodic refractive index perturbation along a length ofthe FBG 20, as shown in operational block 5010. In certain embodimentsof the method 5000, the method 5000 can comprise generating light havinga wavelength, as shown in operational block 5020. The light can begenerated from a narrowband optical source 30. In certain embodiments,the narrowband optical source 30 can be in optical communication with afirst optical path 31 and a second optical path 32. The light can alsosplit into a first portion 33 a of light and a second portion 33 b oflight. The method 5000 further can comprise transmitting a first portion33 a of light along a first optical path 31 extending along and throughthe length of the FBG 20 at a group velocity in operational block 5030.In certain embodiments, the light has a wavelength at or in the vicinityof a wavelength at which the slope of a product of the group index andthe power transmission as a function of wavelength is a maximum (e.g., amaximal value compared to the values of this quantity for any otherwavelengths in the vicinity of an edge of the bandgap of the FBG 20).

Certain embodiments of the method 5000 further can comprise receivingthe first portion 33 a, the second portion 33 b, or both the firstportion 33 a and the second portion 33 b of light with an opticaldetector 40; and detecting an optical power of the first portion 33 a,the second portion 33 b, or both the first portion 33 a and the secondportion 33 b of light. The method 5000 further can comprise reflectingthe second portion 33 b of light along the second optical path 32.

In certain embodiments of the method 5000, the substantially periodicrefractive index perturbation has a constant period along the length ofthe FBG 20. In certain other embodiments, the substantially periodicrefractive index perturbation has a period which varies along the lengthof the FBG 20 such that the FBG 20 is a chirped grating. In someembodiments, the substantially periodic refractive index perturbationhas an amplitude which varies along the length of the FBG 20 such thatthe FBG 20 is an apodized grating.

EXAMPLES

FIG. 24 shows an example experimental setup used to measure thetransmission and group index spectra of an FBG. This optimization usinga figure of merit was reduced to practice by testing an FBG with anominal Δn of 1.035×10⁻³, a length L=1.2 cm, an inferred losscoefficient of 1.16 m⁻¹, and a Gaussian apodized index profile with aFWHM W=0.9 cm. The group delay and the group index were measured asdescribed above in relation to FIG. 24 and Equations 8 and 9.

FIG. 34A shows the measured transmission spectrum and FIG. 34B themeasured group index spectrum of the signal transmitted by the FBG. Thecorresponding solid curves are theoretical predictions calculated with amodel after adjusting Δn, W, and α to best fit them to the experimentalspectra. The fitted values are Δn=1.035×10⁻³, W=0.9 cm, and α=1.16 m⁻¹.The former agrees well with the manufacturer's estimated value. Bothmeasured spectra exhibit multiple ripples that agree well with theory.The maximum measured group index occurs in the vicinity of the secondtransmission peak (λ≈1549.6976 nm) and is equal to 127. The transmissionat this wavelength is 0.8%. To date, this is one of the slowest groupvelocity (2,362 km/s) reported in either an FBG or in an optical fiber.The peak closest to the band edge (λ≈1.54974 μm) should have an evenhigher group index (˜217), but in this example, it could not be measuredbecause the transmitted power at this peak was too small to detect.

To apply a controlled and calibrated strain to this FBG, it was mountedon a piezoelectric (PZT) ring, and an AC voltage was applied to thering, which applied a sinusoidal stretch to the FBG. The PBF was thenplaced in an MZ interferometer, as shown in FIG. 35, to test itsperformance as a slow-light sensor. Care was applied to make sure thatthe two arms of the MZ interferometer had the same length to minimizeconversion of laser phase noise into intensity noise by the MZinterferometer, which would have increased the minimum detectablestrain. The arm length difference, inferred from independentinterferometric measurements, was 1-3 mm. A function generator applied avoltage of known amplitude at frequency ω=25 kHz to the PZT. The outputdetector measured the output of the MZ interferometer. This outputcontains a slowly varying component induced by a slowly varying phasedifference between the two arms due to temperature variations. Thisvariable signal was sent to a proportional-integral-derivative (PID)controller. The latter applied just the right voltage to a second PZTplaced in the lower arm to cancel out this slow drift. The purpose ofthis closed loop was to stabilize the interferometer, and to keep itsphase bias (the phase difference between the two arms, in the absence ofa modulation applied to the FBG/sensing arm) equal to π/2 for maximumsensitivity, as discussed earlier. Another portion of the detectoroutput was sent to a lock-in amplifier, which extracted from it thecomponent of the output that was modulated at ω. This output componentwas the sensor signal. By varying the wavelength of the tunable laser,as in this example, one can measure the response of the sensor to thesame perturbation (strain) amplitude and frequency applied to the FBG asa function of wavelength.

The PZT on which the FBG was attached had been previously calibratedusing a common technique. A known length of fiber was wrapped around it.This fiber was placed inside an MZ interferometer, and the amount ofphase shift occurring in this fiber was measured with thisinterferometer as a function of the voltage applied to the PZT.

Based on the foregoing, the sensitivity of the example sensor of FIG. 35has a spectrum that is proportional to F_(t)(λ)=n_(g)(λ)√T₁(λ), wheren_(g)(λ) is the measured group index spectrum (e.g., FIG. 34B) and T₁(λ)is the transmission spectrum (e.g., FIG. 34A) of the FBG. The spectrumF_(t)(λ) calculated from these two measured spectra can be plotted,e.g., as shown in FIG. 36. As demonstrated earlier in relation to FIG.32, the sensitivity can be expected to be maximum in the vicinity of thepeak j=3 (the first peak, j=1, is not shown because as pointed outearlier it was too weak to be measured). The sensitivity of the sensorto a strain can be defined as

$\begin{matrix}{S = {\frac{1}{P_{0}}\frac{P_{out}}{ɛ}}} & (16)\end{matrix}$

where dP_(out) is the change in power at the output of the MZinterferometer resulting from a change in strain d∈. S is in units ofreciprocal strain. Using Equation 15, it can be seen that thesensitivity is proportional to

$\begin{matrix}{S \propto {2{\eta \left( {1 - \eta} \right)}\sqrt{T_{1}T_{2}}\frac{2\pi}{\lambda}n_{g}}} & (17)\end{matrix}$

where T₂ is the power transmission of the reference arm. Theproportionality factor not shown in Equation 17 depends on the signalprocessing used to extract the ω component of the signal from the outputsignal using well-known art.

The result of the strain-sensing experiment utilizing the setup of FIG.35 is also plotted in FIG. 36 in the form of the sensitivity measured atthe four observed slow light peaks. The measured sensitivities agreereasonably well with the values shown by the spectrum, which again waspredicted from the measured group index and transmission spectra of thegrating. Thus, in certain embodiments, the sensitivity of the slow-lightsensor in accordance with certain embodiments in the transmission modescales like the figure of merit F_(t)(λ)=n_(g)(λ)√T₁(λ), as predicted bytheory. In addition, slow light can play a prominent role in increasingthe sensitivity of the sensor to a strain in certain embodiments. Themaximum measured value shown in FIG. 36, approximately 3.14×10⁵strain⁻¹, is quite possibly the largest ever reported for a strainsensor.

The minimum strain that can be measured with the sensor of FIG. 35 inaccordance with certain embodiments can be calculated from thesensitivity. At the power used in our measurements (a mean detectedpower of P₀=36 μW), the total noise in the output of the sensor(measured in the absence of strain applied to it) ranged from about 1.0μV/√Hz at 3 kHz to about 0.45 μV/√Hz at 30 kHz, or, after calibration, anoise power in the range of P_(noise)≈=25 pW to ˜11 pW. At this outputpower level, this noise was composed of laser relative intensity noise(RIN) and photodetector noise at higher frequencies, as well as dominantlock-in amplifier noise at lower frequencies. Laser phase noise was anegligible component because the MZ interferometer, as mentionedearlier, was very nearly balanced, and phase noise was consequently notconverted into significant intensity noise. The minimum detectablestrain is the strain that produces a variation in output power P_(out)just equal to the noise. From the definition of the sensitivity(Equation 18), the minimum detectable strain (MDS) can be written as:

$\begin{matrix}{ɛ_{m\; i\; n} = {\frac{1}{S}\frac{P_{noise}}{P_{0}}}} & (18)\end{matrix}$

From the measured noise power (25 pW), input power (36 μW), and themaximum sensitivity of 3.14×10⁵ strain⁻¹ measured at the most sensitivepeak (j=3, see, FIG. 36), the MDS is ˜2.2 picostrain (p∈) at 3 kHz and˜1 picostrain at 30 kHz. In comparison, an MDS of 5 p∈ at frequencieslarger than 100 kHz was reported in a π-shifted FBG operated inreflection (D. Gatti, G. Galzerano, D. Janner, S. Longhi, and P.Laporta, “Fiber strain sensor based on a π-phase-shifted Bragg gratingand the Pound-Drever-Hall technique,” Optics Express, Vol. 16, No. 3,1945-50 (Feb. 4, 2008)). Both of these devices were passive. These tworeferences are likely the lowest minimum detectable strain ever reportedprior to the present application for a passive sensor based on an FBG.The Gatti et al. reference does not mention “slow light” at all, and itis mere speculation whether it utilized slow light. In K. P. Koo and A.D. Kersey, “Bragg grating-based laser sensors systems withinterferometric interrogation and wavelength division multiplexing,” J.Lightwave Technol., Vol. 13, No. 7, 1243-49 (July 1995), an MDS of 56femtostrain (or 0.056 p∈) was reported in a fiber laser using an FBG asreflectors. That sensor was an active device that required pump power(˜70 mW) and a strongly imbalanced MZ interferometer (˜100 m of armlength difference), which meant that the MZ interferometer used in thereadout was very difficult to stabilize against external temperaturevariations. In contrast, the device in accordance with certainembodiments described herein used considerably less power (only 36 μW),and its MZ interferometer is nominally balanced, which implies that itis much more thermally stable.

FIG. 37 shows the full width at half maximum bandwidth of the firstslow-light peak predicted from the model, as a function of the indexcontrast, in a uniform grating with a length L=2 cm and substantially noloss. This diagram shows that the bandwidth ranges in this example fromabout 20 pm for a weak grating (index contrast of 10⁻⁴) to about 1 fmfor a strong grating (index contrast of 1.5×10⁻²). At 1.55 μm, the Braggwavelength of this FBG example, for certain embodiments, the linewidthof the laser used to probe the sensor should be lower than thisbandwidth value, and of the order of 1 fm (or lower) to 10 pm (orgreater), depending on the index contrast of the grating. Such lasersare readily available from a large number of manufacturers, such asRedfern Integrated Optics Inc. (RIO) and Agilent Technologies, bothheadquartered in Santa Clara, Calif. This bandwidth depends on a largenumber of parameters, and it can advantageously be either calculated ormeasured experimentally in order to determine the linewidth of the laserto be used in certain embodiments of the sensor. FIG. 37 is shown as anillustration of the process that could be used to determine thislinewidth.

FIG. 38 illustrates the relationship between sensitivity to strain as afunction of index contrast for an FBG with a length L=2 cm and no lossutilized in the slow-light transmission mode (upper solid line), inaccordance with certain embodiments described herein, and theconventional reflection mode (lower solid line) with MZ processing. Asshown in FIG. 38, for the conventional FBG sensor, as the index contrastincreases, the reflection peak broadens and the resolution worsens. Onthe other hand, in both slow-light sensors, sensitivity can increase,e.g., to 30,000 times more sensitive, as the index contrast (and groupindex) increases. Enhancement can be several orders of magnitude foreven modest L, e.g., centimeters, and modest index contrast (e.g.,10⁻³). FIG. 38 illustrates a different numerical example than does FIG.11B, and models strain sensitivity, as opposed to the temperaturesensitivity of FIG. 11B.

FIG. 38 also illustrates the relationship between sensitivity to strainas a function of index contrast for an FBG with a length L=2 cm and withloss utilized in the slow-light transmission mode (dotted and dashedlines), in accordance with certain embodiments described herein. Asshown in FIG. 38, even with loss, e.g., loss in the range of 0.02 m⁻¹ to2 m⁻¹, certain embodiments of sensors described herein can expectsensitivity improvements of 14-1000.

Certain embodiments have demonstrated that very slow light can besupported in FBGs. Group index can increase dramatically with length ofthe FBG, e.g., L^(2.9), and index contrast, e.g., Δn^(1.8). Certainapodized FBGs can provide slower light. In addition, values of 10,000and more have been predicted in low-loss FBGs. As described herein, thelargest group index, e.g., 127 in a 1.2 cm FBG, reported in an opticalfiber, e.g., compare to approximately 5 in silica fiber andapproximately 10 in a Bragg fiber, has been shown. This value of groupindex corresponds to a group velocity as low as 2,360 km/s. Thus,certain embodiments of FBG sensors described herein utilizing slow lighthave enhanced sensitivity with a low minimum detectable strain, e.g.,about 1 p∈ in a passive FBG sensor.

Various embodiments of the present invention have been described above.Although this invention has been described with reference to thesespecific embodiments, the descriptions are intended to be illustrativeof the invention and are not intended to be limiting. Variousmodifications and applications may occur to those skilled in the artwithout departing from the true spirit and scope of the invention asdefined herein.

1. An optical device comprising: a fiber Bragg grating comprising asubstantially periodic refractive index modulation along a length of thefiber Bragg grating, wherein the fiber Bragg grating has a powertransmission spectrum as a function of wavelength comprising a pluralityof local transmission maxima, the local transmission maxima each havinga maximum power at a transmission peak wavelength, the fiber Bragggrating having a group index spectrum as a function of wavelength; and anarrowband optical source in optical communication with a first opticalpath and a second optical path, the narrowband optical source configuredto generate light, the device configured to split the light into a firstportion and a second portion, the first portion transmitted along thefirst optical path extending along and through the length of the fiberBragg grating at a group velocity, wherein the light has a wavelength ator in the vicinity of a wavelength at which one or more of the followingquantities is at a maximum value: (a) the product of the group indexspectrum and a square root of the power transmission spectrum, (b) theslope of a product of the group index spectrum and one minus the powertransmission spectrum, and (c) the slope of a product of the group indexspectrum and the power transmission spectrum.
 2. The optical device ofclaim 1, further comprising at least one optical detector configured toreceive and to detect an optical power of the first portion, the secondportion, or both the first and second portions.
 3. The optical device ofclaim 2, wherein the second portion is transmitted along the secondoptical path, the second optical path not extending along and throughthe fiber Bragg grating.
 4. The optical device of claim 3, furthercomprising: a first fiber coupler in optical communication with thenarrowband optical source, the first optical path, and the secondoptical path; and a second fiber coupler in optical communication withthe at least one optical detector, the first optical path, and thesecond optical path, wherein the at least one optical detector isconfigured to detect a change in power resulting from a change in aphase difference between the first portion and the second portion. 5.The optical device of claim 4, wherein the phase difference isindicative of an amount of strain applied to the fiber Bragg grating. 6.The optical device of claim 4, wherein the phase difference isindicative of a temperature of the fiber Bragg grating.
 7. The opticaldevice of claim 4, wherein the first fiber coupler, second fibercoupler, first optical path, and second optical path form a nominallybalanced Mach-Zehnder interferometer.
 8. The optical device of claim 2,wherein the detected optical power is indicative of an amount of strainapplied to the fiber Bragg grating.
 9. The optical device of claim 2,wherein the detected optical power is indicative of a temperature of thefiber Bragg grating.
 10. The optical device of claim 2, wherein thesecond portion is reflected from the fiber Bragg grating.
 11. Theoptical device of claim 10, wherein the detected optical power isindicative of an amount of strain applied to the fiber Bragg grating.12. The optical device of claim 10, wherein the detected optical poweris indicative of a temperature of the fiber Bragg grating.
 13. Theoptical device of claim 1, wherein the substantially periodic refractiveindex modulation has a constant period along the length of the fiberBragg grating.
 14. A method of using a fiber Bragg grating comprising:providing a fiber Bragg grating comprising a substantially periodicrefractive index modulation along a length of the fiber Bragg grating,wherein the fiber Bragg grating has a power transmission spectrum as afunction of wavelength comprising a plurality of local transmissionmaxima, the local transmission maxima each having a maximum power at atransmission peak wavelength, the fiber Bragg grating having a groupindex spectrum as a function of wavelength; generating light from anarrowband optical source, the narrowband optical source in opticalcommunication with a first optical path and a second optical path,wherein the light is split into a first portion and a second portion;and transmitting the first portion of light along the first optical pathextending along and through the length of the fiber Bragg grating at agroup velocity, wherein the light has a wavelength at or in the vicinityof a wavelength at which one or more of the following quantities is at amaximum value: (a) the product of the group index spectrum and a squareroot of the power transmission spectrum, (b) the slope of a product ofthe group index spectrum and one minus the power transmission spectrum,and (c) the slope of a product of the group index spectrum and the powertransmission spectrum.
 15. The method of claim 14, further comprising:receiving the first portion, the second portion, or both the first andsecond portions with at least one optical detector; and detecting anoptical power of the first portion, the second portion, or both thefirst and second portions.
 16. The method of claim 15, furthercomprising transmitting the second portion along the second opticalpath, the second optical path not extending along and through the fiberBragg grating.
 17. The method of claim 16, further comprising: providinga first fiber coupler in optical communication with the narrowbandoptical source, the first optical path, and the second optical path; andproviding a second fiber coupler in optical communication with theoptical detector, the first optical path, and the second optical path,wherein detecting comprises detecting a phase difference between thefirst portion and the second portion.
 18. The method of claim 17,wherein the phase difference is indicative of an amount of strainapplied to the fiber Bragg grating.
 19. The method of claim 17, whereinthe phase difference is indicative of a temperature of the fiber Bragggrating.
 20. The method of claim 17, wherein the first fiber coupler,second fiber coupler, first optical path, and second optical path form anominally balanced Mach-Zehnder interferometer.
 21. The optical deviceof claim 15, wherein the detected optical power is indicative of anamount of strain applied to the fiber Bragg grating.
 22. The opticaldevice of claim 15, wherein the detected optical power is indicative ofa temperature of the fiber Bragg grating.
 23. The method of claim 15,further comprising reflecting the second portion from the fiber Bragggrating.
 24. The method of claim 23, wherein the detected optical poweris indicative of an amount of strain applied to the fiber Bragg grating.25. The method of claim 23, wherein the detected optical power isindicative of a temperature of the fiber Bragg grating.
 26. The methodof claim 14, wherein the substantially periodic refractive indexmodulation has a constant period along the length of the fiber Bragggrating.
 27. The method of claim 14, wherein the substantially periodicrefractive index modulation has a period that varies along the length ofthe fiber Bragg grating such that the fiber Bragg grating is a chirpedgrating.
 29. The method of claim 17, wherein the substantially periodicrefractive index modulation has an amplitude which varies along thelength of the fiber Bragg grating such that the fiber Bragg grating isan apodized grating.